Consider a circular cone of radius 3 and height 5, which we view horizontally as pictured below. Our goal in this activity is to use a definite integral to determine the volume of the cone.
(a) Find a formula for the linear function \(y = f(x)\) that is pictured above.
\(f(x)=\)
(b) For the representative slice of thickness \(\triangle x\) that is located horizontally at a location \(x\) (somewhere between \(x = 0\) and \(x = 5\)), what is the radius \(r\) of the representative slice? Note that the radius depends on the value of \(x\text{.}\)
\(r =\)
(c) What is the volume \(V_{\small\text{slice}}(x)\) of the representative slice you found in (b)? (Use D as the value for \(\triangle x\) )
\(V_{\small\text{slice}}(x) =\)
(d) What definite integral \(\int_a^b h(x) \ dx\) will sum the volumes of the thin slices across the full horizontal span of the cone?
\(a =\)
\(b =\)
\(h(x) =\)
What is the exact value of this definite integral?
\(\int_a^b h(x) \ dx =\)
(e) Compare the result of your work in (d) to the volume of the cone that comes from using the formula \(V_{\small\text{cone}} = \frac{1}{3} \pi r^2 h.\)
Formula is Larger than the integral
Formula is equal to the integral
Formula is smaller than the integral