Question

Consider the following. y = 2xe−x, y = 0, x = 3; about the y-axis (a) Set up an integral for the volume V of the solid obtained by rotating the region bounded by the given curve about the specified axis. V = 0 dx (b) Use your calculator to evaluate the integral correct to five decimal places.

Answer 1

Using the shell method, the volume would be

$\displaystyle2\pi\int_0^3x(2xe^{-x})\,\mathrm dx=4\pi\int_0^3x^2e^{-x}\,\mathrm dx$

Each shell contributes a volume of approximately 2πrh, where r is the distance from the axis of revolution to the shell (x) and h is the height of the shell given by the vertical distance between y = 2x e^(-x) and y = 0.

If you go on to use your calculator, you'll find the volume is approximately 14.49681.