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 $x$.

$r =$

(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 $V_{\small\text{cone}} = \frac{1}{3} \pi r^2 h.$

You can earn partial credit on this problem.