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Section 6.6 : Work

1. A force of \(F\left( x \right) = {x^2} - \cos \left( {3x} \right) + 2\), \(x\) is in meters, acts on an object. What is the work required to move the object from \(x = 3\) to \(x = 7\)?

Show Solution

There really isn’t all that much to this problem. We are given the force function and limits for the integral (\(x = 3\) and \(x = 7\)) and so all we need to do is write down the integral for the work and evaluate it.

\[\begin{align*}W & = \int_{3}^{7}{{{x^2} - \cos \left( {3x} \right) + 2\,dx}}\\ & = \left. {\left( {\frac{1}{3}{x^3} - \frac{1}{3}\sin \left( {3x} \right) + 2x} \right)} \right|_3^7 = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{1}{3}\left( {340 + \sin \left( 9 \right) - \sin \left( {21} \right)} \right) = 113.1918}}\end{align*}\]