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