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Answers without the blur. Sign up and see all textbooks for free! Q. 35

Expert-verified Found in: Page 511 ### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861 # Consider the region between the graph of $f\left(x\right)=x-2$ and the x-axis on [2,5]. For each line of rotation given in Exercises 35– 40, use definite integrals to find the volume of the resulting solid. The required volume is $36\mathrm{\pi }$

See the step by step solution

## Step 1. Given Information

The given figure is ## Step 2. Calculation

Express the curve as inverse function.

$y=x-2\phantom{\rule{0ex}{0ex}}x=y+2\phantom{\rule{0ex}{0ex}}g\left(y\right)=y+2$

For the x-interval of $\left[2,5\right]$, the corresponding interval of y-variable will be $\left[0,3\right]$

For the washer the external radius of each washer is $x=5$ and internal radius of each wahser is given by $g\left(y\right)$

## Step 3. Calculation

Determine the volume of solid of revolution.

$V=\mathrm{\pi }{\int }_{0}^{3}\left({5}^{2}-{\left(\mathrm{g}\left(\mathrm{y}\right)\right)}^{2}\right)\mathrm{dx}\phantom{\rule{0ex}{0ex}}=\mathrm{\pi }{\int }_{0}^{3}\left(25-{\left(\mathrm{y}+2\right)}^{2}\right)\mathrm{dy}\phantom{\rule{0ex}{0ex}}=\mathrm{\pi }{\int }_{0}^{3}\left(25-{\mathrm{y}}^{2}-4\mathrm{y}-4\right)\mathrm{dy}\phantom{\rule{0ex}{0ex}}=\mathrm{\pi }{\int }_{0}^{3}\left(21-{\mathrm{y}}^{2}-4\mathrm{y}\right)\mathrm{dy}\phantom{\rule{0ex}{0ex}}=\mathrm{\pi }{\left[21\mathrm{y}-\frac{{\mathrm{y}}^{3}}{3}-\frac{4{\mathrm{y}}^{2}}{2}\right]}_{0}^{3}\mathrm{dy}\phantom{\rule{0ex}{0ex}}=\mathrm{\pi }\left(\left(21\left(3\right)-\frac{{3}^{3}}{3}-\frac{4{\left(3\right)}^{2}}{2}\right)-\left(21\left(0\right)-\frac{{0}^{3}}{3}-\frac{4{\left(0\right)}^{2}}{2}\right)\right)\phantom{\rule{0ex}{0ex}}=\mathrm{\pi }\left[\left(63-9-18\right)-0\right]\phantom{\rule{0ex}{0ex}}=36\mathrm{\pi }$ ### Want to see more solutions like these? 