Question

Use the shell method to find the volume of the following solids. A right circular cone of radius 3 and height 8

Short answer

Answer: The volume of the cone is 128π cubic units.

Step-by-step solution

Identify radius and height functions

Let x be the distance from the vertex of the cone along its height. Then, we have to find the radius r(x) and height h(x) as functions of x. Consider a cross-sectional triangle formed by a vertical line segment passing through the cone perpendicular to the base and with vertex at the apex of the cone, height line, and a radius line segment. By similarity, the triangles formed by x and full height are similar. So, r(x)/3 = x/8 => r(x) = 3x/8 h(x) = x (Since the height increases linearly with position x)

Set up the integral for volume

Now, we'll set up the integral representing the volume of the cone using shell method. The formula for the volume of a solid created by revolving function about the x-axis, using shell method is: $$ V = 2\pi \int_{a}^{b} r(x)h(x)dx $$ In our case, we'll integrate from the vertex (x=0) to the base (x=8, since the cone's height is 8). So, a = 0 and b = 8. $$ V = 2\pi \int_{0}^{8} (\frac{3x}{8})(x)dx $$

Solve the integral for the volume

Now, we'll solve the integral to find the volume of the cone. $$ V = 2\pi \int_{0}^{8} (\frac{3x^2}{8})dx $$ $$ V = 2\pi [\frac{3x^3}{24} \Big|_0^8] $$ $$ V = 2\pi [\frac{3(8)^3}{24} - 0] $$ $$ V = 2\pi [\frac{1536}{24}] $$ $$ V = 2\pi [64] $$ $$ V = 128\pi $$ The volume of the right circular cone with radius 3 and height 8 is 128π cubic units.