91Ó°ÊÓ

Finding a Region In Exercises \(7-12\) , the integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral. $$ \int_{-1}^{1}\left[\left(2-x^{2}\right)-x^{2}\right] d x $$

Finding a Region In Exercises \(7-12\) , the integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral. $$ \int_{2}^{3}\left[\left(\frac{x^{3}}{3}-x\right)-\frac{x}{3}\right] d x $$

Finding Arc Length In Exercises \(3-16\) , find the are length of the graph of the function over the indicated interval. $$ y=\frac{x^{5}}{10}+\frac{1}{6 x^{3}}, \quad[2,5] $$

Find the center of mass of the given system of point masses. \(\begin{array}{|c|c|c|c|}\hline m_{i} & {5} & {1} & {3} \\ \hline\left(x_{i}, y_{i}\right) & {(2,2)} & {(-3,1)} & {(1,-4)} \\ \hline\end{array}\)

Hooke's Law In Exercises \(5-10\) , use Hooke's Law to determine the variable force in the spring problem. Eighteen foot-pounds of work is required to stretch a spring 4 inches from its natural length. Find the work required to stretch the spring an additional 3 inches.

Finding the Volume of a Solid In Exercises \(7-10,\) set up and evaluate the integral that gives the volume of the solid formed by revolving the region about the \(y\) -axis. $$ x=-y^{2}+4 y $$

Find the center of mass of the given system of point masses. \(\begin{array}{|c|c|c|c|}\hline m_{i} & {10} & {2} & {5} \\\ \hline\left(x_{i}, y_{i}\right) & {(1,-1)} & {(5,5)} & {(-4,0)} \\\ \hline\end{array}\)

Finding a Region In Exercises \(7-12\) , the integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral. $$ \int_{-\pi / 4}^{\pi / 4}\left(\sec ^{2} x-\cos x\right) d x $$

Finding Arc Length In Exercises \(3-16\) , find the are length of the graph of the function over the indicated interval. $$ y=\frac{3}{2} x^{2 / 3}+4, \quad[1,27] $$

Finding the Volume of a Solid In Exercises \(1-14,\) use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the \(y\) -axis. Graph cannot copy $$ y=x^{3 / 2}, \quad y=8, \quad x=0 $$