🧵 Volume of Solid of Revolution Conceptual Issue

Anonymous at Thu, 20 Feb 2025 03:17:18 UTC No. 16592113

For the area enclosed by the curves y=0, y=8, and y=x^(3/2) we can find a solid of revolution using the shell method with the expected volume being equal to 192pi cubic units. When we integrate in terms of y with the integrand as 2pi y(y^(2/3)dy with upper limit y=8 and lower limit y=0 we find the correct volume. When we integrate with the integrand 2pi x(8-x^3/2)dx with upper limit x=4 and lower limit x=0 we find a value that is off from the correct volume by a multiplication exactly equal to (volume found)(7/2). What is a general method to arrive at the scaling factor for any solid of revolution problem so we may choose to integrate with respect to either variable and find the correct volume?

It may be helpful to notice that when only finding the area under the curve for the integral y=0 to y=8 (y^2/3)dy area found is equal to area found by the integral x=0 to x=4 (8-x^3/2).