91Ó°ÊÓ

Find the centroid of the region bounded by the graphs of the given equations. $$ y=-\frac{2}{3} x+2, \quad y=0, \quad x=0 $$

In Exercises \(9-40\), sketch the region bounded by the graphs of the given equations and find the area of that region. $$ y=x^{2}+3, \quad y=x+1, \quad x=-1, \quad x=1 $$

Prove the identity. \(\sinh ^{2} x=\frac{\cosh 2 x-1}{2}\)

Use the method of cylindrical shells to find the volume of the solid generated by revolving the region bounded by the graphs of the equations and/or inequalities about the indicated axis. Sketch the region and a representative rectangle. \(y=\sqrt{x-1}, \quad y=0, \quad x=5 ;\) the \(y\) -axis

Find the centroid of the region bounded by the graphs of the given equations. $$ y=x^{2}, \quad y=0, \quad x=1, \quad x=2 $$

A force of \(8 \mathrm{lb}\) is required to stretch a spring 2 in. beyond its natural length. Find the work required to stretch the spring 3 in. beyond its natural length.

Find the arc length of the graph of the given equation from \(P\) to \(Q\) or on the specified interval. $$ x=\frac{1}{4} y^{4}+\frac{1}{8 v^{2}} ; \quad P\left(\frac{3}{8}, 1\right), Q\left(\frac{129}{32}, 2\right) $$

In Exercises \(9-40\), sketch the region bounded by the graphs of the given equations and find the area of that region. $$ y=x^{3}+1, \quad y=x-1, \quad x=-1, \quad x=1 $$

Prove the identity. \(\cosh ^{2} x=\frac{1+\cosh 2 x}{2}\)

Use the method of cylindrical shells to find the volume of the solid generated by revolving the region bounded by the graphs of the equations and/or inequalities about the indicated axis. Sketch the region and a representative rectangle. \(y=\frac{1}{x}, \quad y=0, \quad x=1, \quad x=2 ; \quad\) the \(y\) -axis