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APEX Calculus

Section 5.4 The Fundamental Theorem of Calculus

Figure 5.4.1. Video introduction to Section 5.4
Let \(f(t)\) be a continuous function defined on \([a,b]\text{.}\) The definite integral \(\int_a^b f(x)\, dx\) is the “area under \(f\)” on \([a,b]\text{.}\) We can turn this concept into a function by letting the upper (or lower) bound vary.
Let \(F(x) = \int_a^x f(t)\, dt\text{.}\) It computes the area under \(f\) on \([a,x]\) as illustrated in Figure 5.4.2. We can study this function using our knowledge of the definite integral. For instance, \(F(a)=0\) since \(\int_a^af(t)\, dt=0\text{.}\)
A concave-down graph in the first quadrant, with the area underneath shaded from a to x.
The \(t\) and the \(y\) axes are uncalibrated. There are three distinct positions on the \(t\) axis, \(a\text{,}\) \(x\) and \(b\) in the order from left to right. The function \(f(t)\) is a curve facing downward, the areas under the curve between \(a\) and \(x\) are shaded. \(b\) lies to the right of \(x\) outside of the shaded portion.
Figure 5.4.2. The area of the shaded region is \(F(x) = \int_a^x f(t)\, dt\)

Example 5.4.3. Exploring the “Area so far” function.

Consider \(f(t)=2t\) pictured in Figure Figure 5.4.4 and its associated “area so far” function, \(F(x)=\int_1^x 2t\, dt\text{.}\) Using the graph of \(f\) and geometry, find an explicit formula for \(F\text{.}\)
The graph y=2t, and below it, the trapezoidal region from t=1 to t=x is shaded.
The \(y\) axis is drawn from \(-2\) to \(8\) while the \(t\) axis is drawn without calibration but it has two distinct points \(1\) and \(x\text{.}\) The graph is a straight line passing through the origin and has a positive slope. The shaded portion is drawn under the graph between \(1\) and \(x\text{.}\)
Figure 5.4.4. The area of the shaded region is \(F(x) = \int_1^x 2t\, dt\)
Solution 1.
We can see from Figure 5.4.5 that for \(x \geq 1\text{,}\) the area under the curve can be found by subtracting the area of two triangles. The larger triangle will have a base of \(x\) and a height of \(f(x)=2x\text{,}\) while the smaller triangle will have a base of \(1\) and a height of \(2\text{.}\) Therefore, the area under the curve for \(x \geq 1\) is given by \(A(x)=\frac12 (x)(2x)-\frac12 (1)(2)=x^2-1\text{.}\)
The previous trapezoidal region, with heights 2 and 2x shown.
Graph of function same as the one used for the previous example.
The \(y\) axis is drawn from \(-2\) to \(8\) while the \(t\) axis is drawn without calibration but it has two distinct positions \(1\) and \(x\text{.}\) The graph is a straight line passing through the origin and has a positive slope. The shaded portion is drawn under the graph between \(1\) and \(x\text{.}\)
The distance from the origin to point \(x\) on the \(t\) axis is marked \(x\text{.}\) The distance from origin to \(1\) is labeled \(1\text{.}\) The left boundary of the shaded region the \(y\) value is labeled \(2\) and on the right boundaries the \(y\) value is labeled \(2x\text{.}\)
Figure 5.4.5. The area of the shaded region is \(F(x) = \int_1^x 2t\, dt\)
Note that this same formula holds for \(x\lt 1\text{.}\) If \(x \lt 1\text{,}\) then \(F(x) = \int_1^x 2t\, dt=-\int_x^1 2t\, dt\text{.}\) The areas to the left of \(x=1\) will have opposite signs (since they areas are accumulated before \(x=1\)). For example, when \(x=0\text{,}\) \(F(0) = -\int_0^1 2t\, dt=-\frac12 (1)(2)=-1\text{.}\) This is the same value we get from evaluating \(x^2-1\) for \(x=0\text{.}\) Also notice that \(F(-1)=\int_1^{-1} 2t \, dt=-\int_{-1}^1 2t\, dt\text{.}\) This integral is clearly \(0\) since the areas over \([-1,0]\) and \([0,1]\) will sum to zero. Again, this is the same answer obtained by evaluating \(x^2-1\) for \(x=-1\text{.}\)
Therefore, we can reasonably say that \(F(x)=x^2-1\text{.}\) A plot of both \(f(x)=2x\) and \(F(x)=x^2-1\) are given in Figure Figure 5.4.6. You should notice a familiar relationship between these two functions. This relationship is formally stated in Theorem 5.4.7.
Graph of f(x) = 2*x and F(x)= x^2-1.
The \(y\) axis is drawn from \(-2\) to \(8\) and the \(x\) axis is drawn from \(-1\) to \(3\text{.}\) The graphs of two functions are shown. The first is a line through the origin with slope 2, and the second is a parabola that opens upward, with its vertex at \((0,-1)\text{.}\) Both functions intersect at approximately \(x=-0.4\) and \(x=2.4\text{,}\) it is also between these points that the line is above the curve beyond which the curve is above the line.
The straight line that passes through the origin and has a positive slope. It lies in the first and the third quadrant.
The curve appears to start from \(x\) intercept \(1\text{.}\) It has a \(y\) intercept at \(-1\) and the second \(x\) intercept at \(1\text{.}\)
Figure 5.4.6. Graphs of \(f(x)=2x\) and \(F(x)=x^2-1\)
Solution 2. Video solution

Subsection 5.4.1 Fundamental Theorem of Calculus, Parts 1 and 2

As Example 5.4.3 hinted, we can apply calculus ideas to \(F(x)\text{;}\) in particular, we can compute its derivative. In Example 5.4.3, \(F(x)=x^2-1\text{,}\) so \(F'(x)=2x=f(x)\text{.}\) While this may seem like an innocuous thing to do, it has far-reaching implications, as demonstrated by the fact that the result is given as an important theorem.
Figure 5.4.8. Video presentation of Theorem 5.4.7
Initially this seems simple, as demonstrated in the following example.

Example 5.4.9. Using the Fundamental Theorem of Calculus, Part 1.

Let \(\ds F(x) = \int_{-5}^x (t^2+\sin(t) )\, dt\text{.}\) What is \(\Fp(x)\text{?}\)
Solution 1.
Using the Fundamental Theorem of Calculus, we have \(\Fp(x) = x^2+\sin(x)\text{.}\) That is, the derivative of the “area so far” function, is simply the integrand replacing \(x\) with \(t\text{.}\)
This simple example reveals something incredible: \(F(x)\) is an antiderivative of \(x^2+\sin(x)\text{!}\) Therefore, \(F(x) = \frac13x^3-\cos(x) +C\) for some value of \(C\text{.}\) (We can find \(C\text{,}\) but generally we do not care. We know that \(F(-5)=0\text{,}\) which allows us to compute \(C\text{.}\) In this case, \(C=\cos(-5)+\frac{125}3\text{.}\))
Solution 2. Video solution
What we have done in Example 5.4.9 was more than finding a complicated way of computing an antiderivative. Consider a function \(f\) defined on an open interval containing \(a\text{,}\) \(b\) and \(c\text{.}\) Suppose we want to compute \(\int_a^b f(t)\, dt\text{.}\) First, let
\begin{equation} F(x) = \int_c^x f(t)\, dt\text{.}\tag{5.4.1} \end{equation}
Using the properties of the definite integral found in Theorem 5.2.11, we know
\begin{align*} \int_a^b f(t)\, dt \amp = \int_a^c f(t)\, dt + \int_c^b f(t)\, dt\\ \amp = -\int_c^a f(t)\, dt + \int_c^b f(t)\, dt \end{align*}
Using Equation (5.4.1), let \(x=a\) in the first integral and \(x=b\) in the second integral so that \(\int_c^a f(t)\, dt =F(a)\) and \(\int_c^b f(t)\, dt =F(b)\text{.}\) Therefore:
\begin{align*} \int_a^b f(t)\, dt \amp =-F(a) + F(b)\\ \amp = F(b) - F(a)\text{.} \end{align*}
We now see how indefinite integrals and definite integrals are related: we can evaluate a definite integral using antiderivatives! In fact, this is exactly what we noticed in Example 5.4.3. The “area so far” function was indeed an anti-derivative of the integrand. This is the second part of the Fundamental Theorem of Calculus.
Figure 5.4.11. Video presentation of Theorem 5.4.10
As its name suggests, the Fundamental Theorem of Calculus is an important result. In fact, it’s sufficiently important that it’s worth taking a moment to understand why it’s true. A proof is given in Figure 5.4.12.
Figure 5.4.12. Proving the Fundamental Theorem of Calculus

Example 5.4.13. Using the Fundamental Theorem of Calculus, Part 2.

We spent a great deal of time in the previous section studying \(\int_0^4(4x-x^2)\, dx\text{.}\) Using the Fundamental Theorem of Calculus, evaluate this definite integral.
Solution.
We need an antiderivative of \(f(x)=4x-x^2\text{.}\) All antiderivatives of \(f\) have the form \(F(x) = 2x^2-\frac13x^3+C\text{;}\) for simplicity, choose \(C=0\text{.}\)
The Fundamental Theorem of Calculus states
\begin{align*} \int_0^4(4x-x^2)\, dx \amp= F(4)-F(0)\\ \amp = \big(2(4)^2-\frac134^3\big)-\big(0-0\big)\\ \amp = 32-\frac{64}3 = 32/3\text{.} \end{align*}
This is the same answer we obtained using limits in the previous section, just with much less work.
Notation: A special notation is often used in the process of evaluating definite integrals using the Fundamental Theorem of Calculus. Instead of explicitly writing \(F(b)-F(a)\text{,}\) the notation \(F(x)\Big|_a^b\) is used. Thus the solution to Example 5.4.13 would be written as:
\begin{align*} \int_0^4(4x-x^2)\, dx \amp = \left.\left(2x^2-\frac13x^3\right)\right|_0^4\\ \amp = \big(2(4)^2-\frac134^3\big)-\big(0-0\big) = 32/3\text{.} \end{align*}
The Constant \(C\text{:}\) Any antiderivative \(F(x)\) can be chosen when using the Fundamental Theorem of Calculus to evaluate a definite integral, meaning any value of \(C\) can be picked. The constant always cancels out of the expression when evaluating \(F(b)-F(a)\text{,}\) so it does not matter what value is picked. This being the case, we might as well let \(C=0\text{.}\)

Example 5.4.14. Using the Fundamental Theorem of Calculus, Part 2.

Evaluate the following definite integrals.
  1. \(\displaystyle \int_{-2}^2 x^3\, dx\)
  2. \(\displaystyle \int_0^\pi \sin(x) \, dx\)
  3. \(\displaystyle \int_0^5 e^t\, dt\)
  4. \(\displaystyle \int_4^9 \sqrt{u}\, du\)
  5. \(\displaystyle \int_1^5 2\, dx\)
Solution 1.
  1. \begin{align*} \int_{-2}^2 x^3\, dx \amp = \left.\frac14x^4\right|_{-2}^2 \amp\\ \amp = \left(\frac142^4\right) - \left(\frac14(-2)^4\right)\\ \amp = 0\text{.} \end{align*}
  2. \begin{align*} \int_0^\pi \sin(x) \, dx \amp = -\cos(x) \Big|_0^\pi\\ \amp = -\cos(\pi) - \big(-\cos(0) \big)\\ \amp = 1+1=2\text{.} \end{align*}
    (This is interesting; it says that the area under one “hump” of a sine curve is 2.)
  3. \begin{align*} \int_0^5e^t\, dt \amp = e^t\Big|_0^5\\ \amp = e^5 - e^0\\ \amp= e^5-1 \approx 147.41\text{.} \end{align*}
  4. \begin{align*} \int_4^9 \sqrt{u}\, du \amp= \int_4^9 u^\frac12\, du\\ \amp = \frac23u^\frac32\Big|_4^9\\ \amp = \frac23\left(9^\frac32-4^\frac32\right)\\ \amp = \frac23\big(27-8\big) =\frac{38}3\text{.} \end{align*}
  5. \begin{align*} \int_1^5 2\, dx \amp = 2x\Big|_1^5\\ \amp = 2(5)-2\\ \amp =2(5-1)=8\text{.} \end{align*}
    This integral is interesting; the integrand is a constant function, hence we are finding the area of a rectangle with width \((5-1)=4\) and height 2. Notice how the evaluation of the definite integral led to \(2(4)=8\text{.}\) In general, if \(c\) is a constant, then \(\int_a^b c\, dx = c(b-a)\text{.}\)
Solution 2. Video solution

Subsection 5.4.2 Understanding Motion with the Fundamental Theorem of Calculus

We established, starting with Key Idea 2.2.3, that the derivative of a position function is a velocity function, and the derivative of a velocity function is an acceleration function. Now consider definite integrals of velocity and acceleration functions. Specifically, if \(v(t)\) is a velocity function, what does \(\ds \int_a^b v(t)\, dt\) mean?
The Fundamental Theorem of Calculus states that
\begin{equation*} \int_a^b v(t)\, dt = V(b) - V(a)\text{,} \end{equation*}
where \(V(t)\) is any antiderivative of \(v(t)\text{.}\) Since \(v(t)\) is a velocity function, \(V(t)\) must be a position function, and \(V(b) - V(a)\) measures a change in position, or displacement.

Example 5.4.15. Finding displacement and distance.

A ball is thrown straight up with velocity given by \(v(t) = -32t+20\)ft/s, where \(t\) is measured in seconds. Find, and interpret, \(\int_0^1 v(t)\, dt\) and \(\int_0^1 \abs{v(t)}\, dt\text{.}\)
Solution.
Using the Fundamental Theorem of Calculus, we have
\begin{align*} \int_0^1 v(t)\, dt \amp = \int_0^1 (-32t+20)\, dt\\ \amp = \left(-16t^2 + 20t\right)\Big|_0^1\\ \amp = 4\text{.} \end{align*}
Thus if a ball is thrown straight up into the air with velocity \(v(t) = -32t+20\text{,}\) the height of the ball, 1 second later, will be 4 feet above the initial height.
Note that the ball has traveled much farther. It has gone up to its peak and is falling down, but the difference between its height at \(t=0\) and \(t=1\) is 4ft.
If we wish to find the total distance traveled, we must evaluate \(\int_0^1 \abs{v(t)}\, dt\) (noting that negative velocities will reduce the diplacement, but we want distance, not displacement). In this case, we know that the velocity changes sign once when \(v(t)=0\text{,}\)so \(t=20/32=5/8\) seconds. The velocity is positive over \([0,5/8]\) and negative over \([5/8,1]\text{.}\) Therefore
\begin{align*} \int_0^1 \abs{v(t)}\, dt \amp = \int_0^{5/8}v(t)\,dt + \int_{5/8}^1 -v(t)\,dt\\ \amp =\int_0^{5/8}(-32t+20)\, dt-\int_{5/8}^1 (-32t+20)\, dt\\ \amp =\left(-16t^2 + 20t\right)\Big|_0^{5/8}-\left(-16t^2 + 20t\right)\Big|_0^1\\ \amp = \frac{25}4-\left(-\frac94\right)=9\text{.} \end{align*}
So the total distance traveled over \([0,1]\) is \(\int_0^1 \abs{-32t+20}\, dt=9 \text{ feet }\text{.}\)
As we can see in Figure 5.4.16, the positive area between \(v(t)\) and the \(t\)-axis, \(A_1=25/4\text{,}\) while the negative area, \(A_2=-9/4\text{.}\) When we add these two areas, we get the displacement of \(4\) ft. But when we add the absolute value of both of these areas (as in Figure 5.4.17), we get the total distance of \(9\) ft.
Graph showing area between v(t) and the t axis can be used to represent displacement.
The \(y\) axis is drawn from \(-10\) to \(20\) and the \(t\) axis is drawn from \(0\) to \(1\text{.}\) The function is a straight line with a negative slope. The function has an \(y\) intercept of \(20\) and an \(t\) intercept of \(5/8\text{.}\)
The line forms two shaded regions with the \(t\) axis. Between \(0\) to \(5/8\) on the \(t\) axis the region under the graph is shaded and labeled \(A1\text{.}\) The second area is between \(5/8\) and \(1\) that lies in the fourth quadrant and the area is labeled \(A2\text{.}\)
Figure 5.4.16. The area between \(v(t)\) and the \(t\)-axis can be used to represent displacement
Graph showing area between absolute value of v(t) and the t axis can be used to represent displacement.
The \(y\) axis is drawn from \(-10\) to \(20\) and the \(t\) axis is drawn from \(0\) to \(1\text{.}\) The function is a straight line with a negative slope. The function has an \(y\) intercept of \(20 \)and an \(t\) intercept of \(5/8\text{.}\)
The line forms two shaded regions with the \(t\) axis. Between \(0\) to \(5/8\) on the \(t\) axis the region under the graph is shaded and labeled \(A1\text{.}\) Between \(5/8\) and \(1\) lies the second shaded region labeled \(A3\text{.}\)
Figure 5.4.17. The area between \(\abs{v(t)}\) and the \(t\)-axis can be used to represent distance
Integrating a rate of change function gives total change. Velocity is the rate of position change; integrating velocity gives the total change of position, i.e., displacement.
Integrating a speed function gives a similar, though different, result. Speed is also the rate of position change, but does not account for direction. That is, the speed an object is the absolute value of its velocity. This is what we saw in Example 5.4.15 when we evaluated \(\int_0^1 \abs{v(t)}\, dt\text{.}\) So integrating a speed function gives total change of position, without the possibility of “negative position change.” Hence the integral of a speed function gives distance traveled.
As acceleration is the rate of velocity change, integrating an acceleration function gives total change in velocity. We do not have a simple term for this analogous to displacement. If \(a(t) = 5\)miles/h\(^2\) and \(t\) is measured in hours, then
\begin{equation*} \int_0^3 a(t)\, dt = 15 \end{equation*}
means the velocity has increased by 15m/h from \(t=0\) to \(t=3\text{.}\)

Subsection 5.4.3 The Fundamental Theorem of Calculus and the Chain Rule

Part 1 of the Fundamental Theorem of Calculus (FTC) states that given
\begin{equation*} F(x) = \int_a^x f(t)\, dt\text{,} \end{equation*}
we have \(\Fp(x) = f(x)\text{.}\) Using other notation,
\begin{equation*} \lzoo{x}{F(x)}=\lzoo{x}{\int_a^x f(t)\, dt} = f(x)\text{.} \end{equation*}
While we have just practiced evaluating definite integrals, sometimes finding antiderivatives is impossible and we need to rely on other techniques to approximate the value of a definite integral. Functions written as \(F(x) = \int_a^x f(t)\, dt\) are useful in such situations.
It may be of further use to compose such a function with another. As an example, we may compose \(F(x)\) with \(g(x)\) to get
\begin{equation*} F\big(g(x)\big) = \int_a^{g(x)} f(t)\, dt\text{.} \end{equation*}
What is the derivative of such a function? The Chain Rule can be employed to state
\begin{equation*} \frac{d}{dx}\Big(F\big(g(x)\big)\Big) = \Fp\big(g(x)\big)g'(x) = f\big(g(x)\big)g'(x)\text{.} \end{equation*}
Figure 5.4.18. Video presentation of Subsection 5.4.3 and Example 5.4.19
An example will help us understand this.

Example 5.4.19. The FTC, Part 1, and the Chain Rule.

Find the derivative of \(\ds F(x) = \int_2^{x^2} \ln(t) \, dt\text{.}\)
Solution.
We can view \(F(x)\) as being the function \(G(x) = \int_2^x \ln(t) \, dt\) composed with \(g(x) = x^2\text{;}\) that is, \(F(x) = G\big(g(x)\big)\text{.}\) The Fundamental Theorem of Calculus states that \(G'(x) = \ln(x)\text{.}\) The Chain Rule gives us
\begin{align*} F'(x) \amp = G'\big(g(x)\big) g'(x)\\ \amp = \ln(g(x)) g'(x)\\ \amp = \ln(x^2) 2x\\ \amp =2x\ln(x^2) \end{align*}
Normally, the steps defining \(G(x)\) and \(g(x)\) are skipped.
Let’s practice this once more.

Example 5.4.20. The FTC, Part 1, and the Chain Rule.

Find the derivative of \(\ds F(x) = \int_{\cos(x) }^5 t^3\, dt\text{.}\)
Solution 1.
Note that \(\ds F(x) = -\int_5^{\cos(x) } t^3\, dt\text{.}\) Viewed this way, the derivative of \(F\) is straightforward:
\begin{align*} F'(x) \amp= -\cos^3(x)\left(-\sin(x)\right)\\ \amp= \cos^3(x)\sin(x)\text{.} \end{align*}
Solution 2. Video solution

Subsection 5.4.4 Area Between Curves

Figure 5.4.21. Video introduction to Subsection 5.4.4
Consider continuous functions \(f(x)\) and \(g(x)\) defined on \([a,b]\text{,}\) where \(f(x) \geq g(x)\) for all \(x\) in \([a,b]\text{,}\) as demonstrated in Figure 5.4.22. What is the area of the shaded region bounded by the two curves over \([a,b]\text{?}\)
Graph showing area bounded by two functions f(x) and g(x) between x=a and x=b.
The \(x\) and the \(y\) axes are uncalibrated but there are two positions \(a\) and \(b\) marked on the \(x\) axis. There are two functions \(f(x)\) and \(g(x)\) graphed. The graph shows the area that \(f(x)\) forms on \(g(x)\) between positions \(a\) and \(b\text{.}\) The two functions intersect at one point.
The function \(f(x)\) is above \(g(x)\text{,}\) it first curves up and reaches a maxima after which it dips down for a minima and then continues to increase. The function \(g(x)\) has a dip between \(a\) and \(b\) then it increases.
(a)
Graph showing area bounded by two functions f(x) and g(x) between x=a and x=b.
Graph same as previous with additional details. The \(x\) and the \(y\) axes are uncalibrated but there are two positions \(a\) and \(b\) marked on the \(x\) axis. There are two functions \(f(x)\) and \(g(x)\) graphed. The graph shows the area that \(f(x)\) forms on \(g(x)\) between positions \(a\) and \(b\text{.}\) The two functions intersect at one point.
The function \(f(x)\) is above \(g(x)\text{,}\) it first curves up and reaches a maxima after which it dips down for a minima and then continues to increase.The function \(g(x)\) has a dip between \(a\) and \(b\) then it increases. The graph shows areas under both curves as different shaded regions.
(b)
Figure 5.4.22. Finding the area bounded by two functions on an interval by subtracting the area under \(g\) from the area under \(f\)
The area can be found by recognizing that this area is “the area under \(f\) \(-\) the area under \(g\text{.}\)” Using mathematical notation, the area is
\begin{equation*} \int_a^b f(x)\, dx - \int_a^b g(x)\, dx\text{.} \end{equation*}
Properties of the definite integral allow us to simplify this expression to
\begin{equation*} \int_a^b\big(f(x) - g(x)\big)\, dx\text{.} \end{equation*}

Example 5.4.24. Finding area between curves.

Find the area of the region enclosed by \(y=x^2+x-5\) and \(y=3x-2\text{.}\)
Solution 1.
It will help to sketch these two functions, as done in Figure 5.4.25.
Graph of region enclosed by y=x^2+x-5 and y=3*x-2.
The \(y\) axis is drawn from \(-5\) to \(15\) and the \(x\) axis is drawn from \(-2\) to \(4\text{.}\) The first function \(x^2 +x-5\)is a curve that has a \(y\) intercept at \(-5\) and the \(x\) intercept near \(1.8\text{.}\) The second function \(3*x-2\) is a straight line. The two lines intersect at points \((-1,-5)\) and \((3,7)\text{.}\) The area bounded by the two functions is shaded.
Figure 5.4.25. Sketching the region enclosed by \(y=x^2+x-5\) and \(y=3x-2\) in Example 5.4.24
The region whose area we seek is completely bounded by these two functions; they seem to intersect at \(x=-1\) and \(x=3\text{.}\) To check, set \(x^2+x-5=3x-2\) and solve for \(x\text{:}\)
\begin{align*} x^2+x-5 \amp = 3x-2\\ (x^2+x-5) - (3x-2) \amp = 0\\ x^2-2x-3 \amp = 0\\ (x-3)(x+1) \amp = 0\\ x\amp =-1,\,3\text{.} \end{align*}
Following Theorem 5.4.23, the area is
\begin{align*} \int_{-1}^3\big(3x-2 -(x^2+x-5)\big)\, dx \amp = \int_{-1}^3 (-x^2+2x+3)\, dx\\ \amp =\left.\left(-\frac13x^3+x^2+3x\right)\right|_{-1}^3\\ \amp =-\frac13(27)+9+9-\left(\frac13+1-3\right)\\ \amp = 10\frac23 = 10.\overline{6} \end{align*}
Solution 2. Video solution
One of the things we have to be careful about when finding the area between curves is that the curves might cross, so that the distinction between “upper curve” and “lower curve” can change. The video example in Figure 5.4.26 illustrates this phenomenon.
Figure 5.4.26. Finding the area between curves that intersect multiple times

Subsection 5.4.5 The Mean Value Theorem and Average Value

A graph of a function to introduce the Mean Value Theorem.
The \(y\) axis is uncalibrated and the \(x\) axis is drawn from \(0\) to \(4\text{.}\) The curve first decreases from \(0\) to \(3\) then it increases until \(x=4\text{.}\)
Figure 5.4.27. A graph of a function \(f\) to introduce the Mean Value Theorem
Consider the graph of a function \(f\) in Figure 5.4.27 and the area defined by \(\int_1^4 f(x)\, dx\text{.}\) Three rectangles are drawn in Figure 5.4.28; in Figure 5.4.28.(a), the height of the rectangle is greater than \(f\) on \([1,4]\text{,}\) hence the area of this rectangle is is greater than \(\int_1^4 f(x)\, dx\text{.}\)
In Figure 5.4.28.(b), the height of the rectangle is smaller than \(f\) on \([1,4]\text{,}\) hence the area of this rectangle is less than \(\int_1^4 f(x)\, dx\text{.}\)
Finally, in Figure 5.4.28.(c) the height of the rectangle is such that the area of the rectangle is exactly that of \(\int_1^4 f(x)\, dx\text{.}\) Since rectangles that are “too big”, as in (a), and rectangles that are “too little,” as in (b), give areas greater/lesser than \(\int_1^4 f(x)\, dx\text{,}\) it makes sense that there is a rectangle, whose top intersects \(f(x)\) somewhere on \([1,4]\text{,}\) whose area is exactly that of the definite integral.
Graph of function with rectangle to show upper bound on function.
The \(y\) axis is uncalibrated and the \(x\) axis is drawn from \(0\) to \(4\text{.}\) The curve first decreases from \(0\) to \(3\) then it increases until \(x=4\text{.}\) The area under the curve between \(x=1\) to \(x=4\) is shaded. A rectangular boundary is drawn on the \(x\) axis with height above the curve.
(a)
Graph of function with rectangle to show lower bound on function.
The \(y\) axis is uncalibrated and the \(x\) axis is drawn from \(0\) to \(4\text{.}\) The curve first decreases from \(0\) to \(3\) then it increases until \(x=4\text{.}\) The area under the curve between \(x=1\) to \(x=4\) is shaded. A rectangular boundary is drawn on the \(x\) axis with height a little below the curve.
(b)
Graph of function with rectangle that matches area of function exactly.
The \(y\) axis is uncalibrated and the \(x\) axis is drawn from \(0\) to \(4\text{.}\) The curve first decreases from \(0\) to \(3\) then it increases until \(x=4\text{.}\) The area under the curve between \(x=1\) to \(x=4\) is shaded. A rectangular boundary is drawn on the \(x\) axis such that from \(1\) to \(2\) the curve is above the rectangle and from \(2\) to \(4\) the curve is below the rectangle.
(c)
Figure 5.4.28. Differently sized rectangles give upper and lower bounds on \(\int_1^4 f(x)\, dx\text{;}\) the last rectangle matches the area exactly
We state this idea formally in a theorem.
Figure 5.4.30. Video presentation of Theorem 5.4.29
This is an existential statement; \(c\) exists, but we do not provide a method of finding it. Theorem 5.4.29 is directly connected to the Mean Value Theorem of Differentiation, given as Theorem 3.2.4; we leave it to the reader to see how.
We demonstrate the principles involved in this version of the Mean Value Theorem in the following example.

Example 5.4.31. Using the Mean Value Theorem.

Consider \(\int_0^\pi \sin(x) \, dx\text{.}\) Find a value \(c\) guaranteed by the Mean Value Theorem.
Solution.
We first need to evaluate \(\int_0^\pi \sin(x) \, dx\text{.}\) (This was previously done in Example 5.4.14.)
\begin{equation*} \int_0^\pi\sin(x) \, dx = -\cos(x) \Big|_0^\pi = 2\text{.} \end{equation*}
Thus we seek a value \(c\) in \([0,\pi]\) such that \(\pi\sin(c) =2\text{.}\)
\begin{equation*} \pi\sin(c) = 2\,\,\Rightarrow\,\,\sin(c) = 2/\pi\,\,\Rightarrow\,\,c = \arcsin(2/\pi) \approx 0.69\text{.} \end{equation*}
Graph of function y=sin(x) and the rectangle guaranteed by the Mean Value Theorem.
The \(y\) axis has two positions marked \(sin(0.69)\) and \(1\text{,}\) the \(x\) axis has \(c\text{,}\) \(1\text{,}\) \(2\) and \(\pi\) marked in order. The function is an inverted parabola of height \(1\) and has \(x\) intercepts at \(x=0\) and \(x=\pi\text{.}\) The area under the parabola until the \(x\) axis is shaded. A rectangular boundary is drawn with height \(sin(0.69)\) and width \(\pi\)
Figure 5.4.32. A graph of \(y=\sin(x)\) on \([0,\pi]\) and the rectangle guaranteed by the Mean Value Theorem
In Figure 5.4.32 \(\sin(x)\) is sketched along with a rectangle with height \(\sin(0.69)\text{.}\) The area of the rectangle is the same as the area under \(\sin(x)\) on \([0,\pi]\text{.}\)
We now turn our attention to a related topic —average value. Let \(f\) be a function on \([a,b]\) with \(c\) such that \(f(c)(b-a) = \int_a^bf(x)\, dx\text{.}\) Consider \(\int_a^b\big(f(x)-f(c)\big)\, dx\text{:}\)
\begin{align*} \int_a^b\big(f(x)-f(c)\big)\, dx \amp = \int_a^b f(x) - \int_a^b f(c)\, dx\\ \amp = f(c)(b-a) - f(c)(b-a)\\ \amp = 0\text{.} \end{align*}
When \(f(x)\) is shifted by \(-f(c)\text{,}\) the amount of area under \(f\) above the \(x\)-axis on \([a,b]\) is the same as the amount of area below the \(x\)-axis above \(f\text{;}\) see Figure 5.4.33 for an illustration of this. In this sense, we can say that \(f(c)\) is the average value of \(f\) on \([a,b]\text{.}\)
A graph of y = f(x) and the rectangle guaranteed by the Mean Value Theorem.
The \(x\) and the \(y\) axes are uncalibrated. The \(y\) axis is labeled as \(f(x)\text{.}\) The \(x\) axis has three distinct positions marked \(a\text{,}\) \(c\) and \(b\) in the order.
The function represents a parabola and the graph shows the area under the parabola. \(a\) and \(b\) are the two ends for the area. \(c\) is to the left of the vertex of the parabola almost at half the distance from \(a\text{.}\)
The rectangle guaranteed by the Mean Value Theorem lies above the vertex of the parabola.
Graph of f(x) is shifted down by f(c), the resulting “area under the curve” is 0.
The \(x\) and the \(y\) axes are uncalibrated. The \(y\) axis is labeled as \(f(x)\text{.}\) The \(x\) axis has three distinct positions marked \(a\text{,}\) \(c\) and \(b\) in the order.
The function represents a parabola and the graph shows the area under the parabola. \(a\) and \(b\) are the two ends for the area. \(c\) is to the left of the vertex of the parabola almost at half the distance from \(a\text{.}\)
Sine the function is shifted by \(f(c)\text{,}\) there are three parts to the area, two are positive and are above the \(x\) axis, from \(a\) to \(c\) and from a little before \(b\) to \(b\text{.}\) From a little before \(b\) to \(b\) the area is drawn in the fourth quadrant from above the parabola on the \(x\) axis.
Figure 5.4.33. On the left, a graph of \(y=f(x)\) and the rectangle guaranteed by the Mean Value Theorem. On the right, \(y=f(x)\) is shifted down by \(f(c)\text{;}\) the resulting “area under the curve” is 0
The value \(f(c)\) is the average value in another sense. First, recognize that the Mean Value Theorem can be rewritten as
\begin{equation*} f(c) = \frac{1}{b-a}\int_a^b f(x)\, dx\text{,} \end{equation*}
for some value of \(c\) in \([a,b]\text{.}\) Replacing the integral with the limit of a Riemann sum (as in Theorem 5.3.26):
\begin{align*} f(c) \amp = \frac{1}{b-a}\int_a^b f(x)\, dx \amp\\ \amp = \frac{1}{b-a} \lim\limits_{n \to \infty}\sum_{i=1}^n f(c_i)\,\Delta x \amp \text{Using } \knowl{./knowl/xref/thm_riemann_sum.html}{\text{Theorem 5.3.26}}\\ \amp =\frac{1}{b-a} \lim\limits_{n \to \infty}\sum_{i=1}^n f(c_i)\,\frac{b-a}{n}\amp \Delta x =\frac{b-a}{n}\\ \amp =\lim\limits_{n \to \infty}\sum_{i=1}^n f(c_i)\frac{1}{n} \amp \text{Cancelling the common factor of } b-a\text{.} \end{align*}
Examining this last line closely, the expression \(\sum_{i=1}^n f(c_i)\frac{1}{n}\) represents adding up \(n\) sample values of \(f(x)\)and then dividing by \(n\text{.}\) This is exactly what we do when we calculate the average of a set of \(n\) numbers. Now when we consider taking the limit as \(n\) goes to \(\infty\text{,}\) \(\lim\limits_{n \to \infty}\sum_{i=1}^n f(c_i)\frac{1}{n}\text{,}\) we are adding up all of the function’s output values over \([a,b]\) and dividing by the “number of numbers”. In a sense, we are adding up an infinite number of output values and then dividing by the number of terms we summed (which is again infinite).
This leads us to a definition.

Definition 5.4.34. The Average Value of \(f\) on \([a,b]\).

Let \(f\) be continuous on \([a,b]\text{.}\) The average value of \(f\) on \([a,b]\) is \(f(c)\text{,}\) where \(c\) is a value in \([a,b]\) guaranteed by the Mean Value Theorem. i.e.,
\begin{equation*} \text{ Average Value of \(f\) on \([a,b]\) } = \frac{1}{b-a}\int_a^b f(x)\, dx\text{.} \end{equation*}
Figure 5.4.35. Video presentation of Definition 5.4.34
An application of this definition is given in the following example.

Example 5.4.36. Finding the average value of a function.

An object moves back and forth along a straight line with a velocity given by \(v(t) = (t-1)^2\) on \([0,3]\text{,}\) where \(t\) is measured in seconds and \(v(t)\) is measured in ft/s.
What is the average velocity of the object?
Solution.
By our definition, the average velocity is:
\begin{align*} \frac{1}{3-0}\int_0^3 (t-1)^2\, dt \amp =\frac13 \int_0^3 \big(t^2-2t+1\big)\, dt\\ \amp = \left.\frac13\left(\frac13t^3-t^2+t\right)\right|_0^3\\ \amp =\frac13\left[\left(\frac13(3)^3-(3)^2+(3)\right)-\left(\frac13(0)^3-(0)^2+(0)\right)\right]\\ \amp = 1\text{ ft/s }\text{.} \end{align*}
We can understand the above example through a simpler situation. Suppose you drove 100 miles in 2 hours. What was your average speed? The answer is simple: displacement/time = 100 miles/2 hours = 50 mph.
What was the displacement of the object in Example 5.4.36? We calculate this by integrating its velocity function: \(\int_0^3 (t-1)^2\, dt = 3\) ft. Its final position was 3 feet from its initial position after 3 seconds: its average velocity was 1 ft/s.
This section has laid the groundwork for a lot of great mathematics to follow. The most important lesson is this: definite integrals can be evaluated using antiderivatives. Since Section 5.3 established that definite integrals are the limit of Riemann sums, we can later create Riemann sums to approximate values other than “area under the curve,” convert the sums to definite integrals, then evaluate these using the Theorem 5.4.10. This will allow us to compute the work done by a variable force, the volume of certain solids, the arc length of curves, and more.
The downside is this: generally speaking, computing antiderivatives is much more difficult than computing derivatives. Chapter 6 is devoted to techniques of finding antiderivatives so that a wide variety of definite integrals can be evaluated. Before that, Section 5.5 explores techniques of approximating the value of definite integrals beyond using the Left Hand, Right Hand and Midpoint Rules. These techniques are invaluable when antiderivatives cannot be computed, or when the actual function \(f\) is unknown and all we know is the value of \(f\) at certain \(x\)-values.

Exercises 5.4.6 Exercises

Terms and Concepts

1.
How are definite and indefinite integrals related?
2.
What constant of integration is most commonly used when evaluating definite integrals?
3.
  • True
  • False
If \(f\) is a continuous function, then \(\ds F(x) = \int_a^x f(t)\, dt\) is also a continuous function.
4.
The definite integral can be used to find “the area under a curve.” Give two other uses for definite integrals.

Problems

Exercise Group.
Evaluate the definite integral.
5.
\(\ds\int_{2}^{3} \left({3x^{2}+2x-8}\right)\, dx\)
6.
\(\ds\int_{0}^{5} {\left(x-8\right)^{2}}\, dx\)
7.
\(\ds\int_{-3}^{3} \left({x^{5}-x^{7}}\right)\, dx\)
8.
\(\ds\int_{{0}}^{{\frac{\pi }{2}}} {\sin\mathopen{}\left(x\right)}\, dx\)
9.
\(\ds\int_{{\frac{\pi }{4}}}^{{\frac{\pi }{3}}} {\sec^{2}\mathopen{}\left(x\right)}\, dx\)
10.
\(\ds\int_{1}^{{e^{5}}} {\frac{1}{x}}\, dx\)
11.
\(\ds\int_{-3}^{1} {6^{x}}\, dx\)
12.
\(\ds\int_{-3}^{-2} \left({4-7x^{3}}\right)\, dx\)
13.
\(\ds\int_{0}^{\pi} \left({6\cos\mathopen{}\left(x\right)-4\sin\mathopen{}\left(x\right)}\right)\, dx\)
14.
\(\ds\int_{3}^{4} {e^{x}}\, dx\)
15.
\(\ds\int_{0}^{16} {\sqrt{t}}\, dt\)
16.
\(\ds\int_{1}^{9} {\frac{1}{\sqrt{t}}}\, dt\)
17.
\(\ds\int_{27}^{64} {\sqrt[3]{x}}\, dx\)
18.
\(\ds\int_{1}^{5} {\frac{1}{x}}\, dx\)
19.
\(\ds\int_{1}^{5} {\frac{1}{x^{2}}}\, dx\)
20.
\(\ds\int_{1}^{9} {\frac{1}{x^{5}}}\, dx\)
21.
\(\ds\int_{0}^{1} {x}\, dx\)
22.
\(\ds\int_{0}^{1} {x^{2}}\, dx\)
23.
\(\ds\int_{0}^{1} {x^{3}}\, dx\)
24.
\(\ds\int_{0}^{1} {x^{81}}\, dx\)
25.
\(\ds\int_{-8}^{8} dx\)
26.
\(\ds\int_{-7}^{-2} {6}\, dx\)
27.
\(\ds\int_{-6}^{6} {0}\, dx\)
28.
\(\ds\int_{{\frac{\pi }{6}}}^{{\frac{\pi }{4}}} {\csc^{2}\mathopen{}\left(x\right)}\, dx\)
29.
(a)
Explain why \(\ds\int_{-1}^1 x^n\, dx=0\) when \(n\) is a positive, odd integer.
(b)
Explain why \(\ds\int_{-1}^1 x^n\, dx = 2\int_{0}^1 x^n\, dx\) when \(n\) is a positive, even integer.
30.
Explain why \(\ds\int_{a}^{a+2\pi} \sin t\, dt = 0\) for all values of \(a\text{.}\)
Exercise Group.
Find all values \(c\) such that \(\int_a^b f(x)\, dx = f(c)(b-a)\text{,}\) as guaranteed by the Theorem 5.4.29.
31.
\(\int_{}^{}{x^{2}} \, dx\)
32.
\(\int_{}^{}{x^{2}} \, dx\)
33.
\(\int_{}^{}{e^{x}} \, dx\)
34.
\(\int_{}^{}{\sqrt{x}} \, dx\)
Exercise Group.
Find the average value of the function on the given interval.
35.
\(f(x)={\sin\mathopen{}\left(x\right)}\) on \(\left[{\frac{\pi }{2}},{\pi }\right]\)
36.
\(y={\sin\mathopen{}\left(x\right)}\) on \([{0},{\frac{\pi }{2}}]\)
37.
\(y={x}\) on \([0,5]\)
38.
\(y={x^{2}}\) on \([0,6]\)
39.
\(y={x^{3}}\) on \([0,7]\)
40.
\(y=\ds{\frac{1}{t}}\) on \(\left[1,{e^{8}}\right]\)
Exercise Group.
A velocity function is given for an object moving along a straight line. Find the displacement of the object over the given time interval.
41.
\(v(t)={-32t+22\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}}\) on \([0,8]\)
42.
\(v(t)={-32t+140\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}}\) on \([0,7]\)
43.
\(v(t)={19\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}}\) on \([0,3]\)
44.
\(v(t)={2^{t}\ {\rm mph}}\) on \([-3,2]\)
45.
\(v(t)={\sin\mathopen{}\left(t\right)\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}}\) on \(\left[{0},{\pi }\right]\)
46.
\(v(t)={\sqrt[4]{t}\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}}\) on \([0,256]\)
Exercise Group.
An acceleration function of an object moving along a straight line is given. Find the change of the object’s velocity over the given time interval.
47.
\(a(t)={-32\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s^{2}}}}\) on \([0,6]\)
48.
\(a(t)={7\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s^{2}}}}\) on \([0,7]\)
49.
\(a(t)={t\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s^{2}}}}\) on \([0,8]\)
50.
\(a(t)={\cos\mathopen{}\left(t\right)\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s^{2}}}}\) on \(\left[{0},{\frac{3\pi }{2}}\right]\)
Exercise Group.
Sketch the given relations and find the area of the enclosed region.
51.
\(y=2x\text{,}\) \(y=5x\text{,}\) and \(x= 3\)
52.
\(y=-x+1\text{,}\) \(y=3x+6\text{,}\) \(x=2\) and \(x= -1\)
53.
\(y=x^2-2x+5\text{,}\) \(y=5x-5\)
54.
\(y=2x^2+2x-5\text{,}\) \(y=x^2+3x+7\text{,}\)
Exercise Group.
Find \(F'(x)\text{.}\)
55.
\(F(x) = \ds\int_{9}^{{x^{3}-7x}} {\frac{1}{t}}\, dt\)
56.
\(F(x) = \ds\int_{{x^{2}}}^{5} {t^{2}}\, dt\)
57.
\(F(x) = \ds\int_{{x}}^{{x^{3}}} ({t-5})\, dt\)
58.
\(F(x) = \ds\int_{{\ln\mathopen{}\left(x\right)}}^{{e^{x}}} {\sin\mathopen{}\left(t\right)}\, dt\)
59.
\(F(x) = \ds\int_{3}^{{x^{3}}} ({\sin\mathopen{}\left(4t^{2}\right)})\, dt\)
60.
\(F(x) = \ds\int_{{\ln\mathopen{}\left(x\right)}}^{{e^{x}}} ({\sqrt{t^{4}+2t^{2}}})\, dt\)
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