## y = 5xe−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

Question

y = 5xe−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.
(b) Use your calculator to evaluate the integral correct to five decimal places.

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5 months 2021-12-27T09:03:20+00:00 1 Answer 0 views 0

1. Using the shell method, the volume is

$$V=\displaystyle2\pi\int_0^3x\cdot5xe^{-x}\,\mathrm dx=10\pi\int_0^3x^2e^{-x}\,\mathrm dx$$

Integrate by parts, taking

$$u=x^2\implies\mathrm du=2x\,\mathrm dx$$

$$\mathrm dv=e^{-x}\,\mathrm dx\implies v=-e^{-x}$$

$$V=\displaystyle10\pi\left(-x^2e^{-x}\bigg|_0^3+2\int_0^3xe^{-x}\,\mathrm dx\right)$$

$$V=\displaystyle-90\pi e^{-3}+20\pi\int_0^3xe^{-x}\,\mathrm dx$$

Integrate by parts again, with

$$u=x\implies\mathrm du=\mathrm dx$$

$$\mathrm dv=e^{-x}\,\mathrm dx\implies v=-e^{-x}$$

$$V=-90\pi e^{-3}+20\pi\left(-xe^{-x}\bigg|_0^3\displaystyle\int_0^3e^{-x}\,\mathrm dx\right)$$

$$V=-150\pi e^{-3}+20\pi\displaystyle\int_0^3e^{-x}\,\mathrm dx$$

$$V=\displaystyle-150\pi e^{-3}+20\pi(-e^{-x})\bigg|_0^3$$

$$\boxed{V=20\pi-170\pi e^{-3}}$$