Answer to Question #237915 in Calculus for sayem

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.

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

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

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

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

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

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