Short Answer

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

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

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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 $x = \frac{π}{2}$ if the force acting on the object at a given value of x is data-custom-editor="chemistry" $F (x) = xsinx$ .

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

$W = \int_{0}^{π / 2} x \sin x d x$

Firstly solving the integral

$W = \int x \sin x d x W = [x \int \sin x d x - \{\int \frac{d}{d x} (x) (\int \sin x d x)\}] W = [- x \cos x - \{- \frac{x^{2}}{2} \cos x\}] W = [- x \cos x + \frac{x^{2}}{2} \cos x]$

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

$W = {[- x \cos x + \frac{x^{2}}{2} \cos x]}_{0}^{π / 2} W = [(- \frac{π}{2} \cos \frac{π}{2} + \frac{{(\frac{π}{2})}^{2}}{2} \cos \frac{π}{2}) - (0 \cos 0 + \frac{{(0)}^{2}}{2} \cos 0)] W = [(- \frac{π}{2} \times 0 + \frac{π^{2}}{4 \cdot 2} \times 0) - (0 \cos 0 + \frac{{(0)}^{2}}{2} \cos 0)] W = [(0 + 0) - (0 + 0)] W = 0$

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Short Answer

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

Step by Step Solution

Step 1.Given Information

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

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

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Calculus

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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=π2 if the force acting on the object at a given value of x is role="math" localid="1650297715748" F(x)=xsinx.

Step by Step Solution

Step 1.Given Information

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

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

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