Question

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

1. y=ln(x), y=1, y=2, x=0; about the y axis
2. y=e^(-x), y=1, x=2; about y=2

Answer 1

1.

$y=\ln x=>x=e^y$

$$V=\pi\displaystyle\int_{1}^{2}(e^y)^2dy=\pi[\dfrac{e^{2y}}{2}]\begin{matrix} 2\\ 1 \end{matrix}$$

$$=\dfrac{\pi(e^4-e^2)}{2}({units}^3)$$

2.

$$e^{-x}=1=>x=0$$

$$V=\pi\displaystyle\int_{0}^{2}[(2-e^{-x})^2-(2-1)^2]dx$$

$$=\pi\displaystyle\int_{0}^{2}[3-4e^{-x}+e^{-2x}]dx$$

$$=\pi[3x+4e^{-x}-\dfrac{e^{-2x}}{2}]\begin{matrix} 2\\ 0 \end{matrix}$$

$$=\pi(6+4e^{-2}-\dfrac{e^{-4}}{2}-0-4+\dfrac{1}{2})$$

$$=\pi(\dfrac{5}{2}+4e^{-2}-\dfrac{e^{-4}}{2})({units}^3)$$