Question

Find the volume generated by revolving the area bounded by y=e^x, x=1, x=0, about y-axis. Sketch the curve indicating the generating area.

Answer 1

$$y=e^x, x=0, x=1$$

$$V=\displaystyle\int_{0}^12\pi x(e^x)dx$$

$\displaystyle\int xe^xdx$

$$\int udv=uv-\int vdu$$

$u=x, du=dx$

$dv=e^xdx, v=\displaystyle\int e^xdx=e^x$

$$\displaystyle\int xe^xdx=xe^x-\displaystyle\int e^xdx=xe^x-e^x+C$$

$$V=\displaystyle\int_{0}^12\pi x(e^x)dx=2\pi\big[xe^x-e^x\big]\begin{matrix} 1 \\ 0 \end{matrix}$$

$$=2\pi(e-e-(0-1))=2\pi (units^3)$$

$V=2\pi\ cubic\ units$