Volume of Revolution: Cone

Just as we can use definite integrals to add the areas of rectangular slices to find the exact area that lies between two curves, we can also use integrals to find the volume of regions whose cross-sections have a particular shape.[br][br]In particular, we can determine the volume of solids whose cross-sections are all thin cylinders (or washers) by adding up the volumes of these individual slices. We first consider a familiar shape: a circular cone.

[size=150]Consider a circular cone of radius 3 and height 5, which we view horizontally as pictured above. Our goal in this activity is to use a definite integral to determine the volume of the cone.[/size]

Question 1

Find a formula for the linear function [math]y=f\left(x\right)[/math] that is pictured in the figure.

Question 2

For the representative slice of thickness Δ𝑥 that is located horizontally at a location 𝑥 (somewhere between 𝑥 = 0 and 𝑥 = 5), what is the radius of the representative slice? [br][br]Note that the radius depends on the value of 𝑥.

Question 3

What is the volume of the representative slice you found in Question 2?[br]

Question 4

What definite integral will sum the volumes of the thin slices across the full horizontal span of the cone? What is the exact value of this definite integral?[br]

Question 5

Compare the result of your work in (d) to the volume of the cone that comes from using the formula [math]V=\frac{1}{3}\pi r^2h[/math]

[size=200][size=150]How can we use this strategy to find the volume of any cone with radius = r and height = h?[/size][/size]