Problem 27 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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)

Step 2: 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. $$ V = 2\pi \int_{0}^{8} (\frac{3x}{8})(x)dx $$

Step 3: 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.