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Q. 1

Expert-verified
Found in: Page 1095

Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

Work as an integral of force and distance: Find the work done in moving an object along the x-axis from the origin to $x=\frac{\mathrm{\pi }}{2}$ if the force acting on the object at a given value of x is role="math" localid="1650297715748" $\mathrm{F}\left(x\right)=x\mathrm{sin}x$.

The work done in moving an object along the x-axis from the origin to$x=\frac{\mathrm{\pi }}{2}$ is $0$.

See the step by step solution

Step 1.Given Information

Work as an integral of force and distance: Find the work done in moving an object along the x-axis from the origin to $x=\frac{\mathrm{\pi }}{2}$ if the force acting on the object at a given value of x is data-custom-editor="chemistry" $\mathrm{F}\left(\mathrm{x}\right)=\mathrm{xsinx}$.

Step 2. The work done is W=∫0π/2F(x)dx

$W={\int }_{0}^{\mathrm{\pi }/2}x\mathrm{sin}xdx$

Firstly solving the integral

$W=\int x\mathrm{sin}xdx\phantom{\rule{0ex}{0ex}}W=\left[x\int \mathrm{sin}x\mathrm{d}x-\left\{\int \frac{d}{dx}\left(x\right)\left(\int \mathrm{sin}xdx\right)\right\}\right]\phantom{\rule{0ex}{0ex}}W=\left[-x\mathrm{cos}x-\left\{-\frac{{x}^{2}}{2}\mathrm{cos}x\right\}\right]\phantom{\rule{0ex}{0ex}}W=\left[-x\mathrm{cos}x+\frac{{x}^{2}}{2}\mathrm{cos}x\right]$

Step 3. Now solving the integral W=-xcosx+x22cosx0π/2

$W={\left[-x\mathrm{cos}x+\frac{{x}^{2}}{2}\mathrm{cos}x\right]}_{0}^{\mathrm{\pi }/2}\phantom{\rule{0ex}{0ex}}W=\left[\left(-\frac{\mathrm{\pi }}{2}\mathrm{cos}\frac{\mathrm{\pi }}{2}+\frac{{\left(\frac{\mathrm{\pi }}{2}\right)}^{2}}{2}\mathrm{cos}\frac{\mathrm{\pi }}{2}\right)-\left(0\mathrm{cos}0+\frac{{\left(0\right)}^{2}}{2}\mathrm{cos}0\right)\right]\phantom{\rule{0ex}{0ex}}W=\left[\left(-\frac{\mathrm{\pi }}{2}×0+\frac{{\mathrm{\pi }}^{2}}{4·2}×0\right)-\left(0\mathrm{cos}0+\frac{{\left(0\right)}^{2}}{2}\mathrm{cos}0\right)\right]\phantom{\rule{0ex}{0ex}}W=\left[\left(0+0\right)-\left(0+0\right)\right]\phantom{\rule{0ex}{0ex}}W=0$