91Ó°ÊÓ

In Exercises 29-32, use the disk method or the shell method to find the volumes of the solids generated by revolving the region bounded by the graphs of the equations about the given lines. $$\begin{array}{l}{x^{2 / 3}+y^{2 / 3}=a^{2 / 3}, \quad a>0 \text { (hypocycloid) }} \\ {\text { (a) the } x \text { -axis } \quad \text { (b) the } y \text { -axis }}\end{array}$$

Length of a Cable An electric cable is hung between two towers that are 40 meters apart (see figure). The cable takes the shape of a catenary whose equation is \(y=10\left(e^{x / 20}+e^{-x / 20}\right), \quad-20 \leq x \leq 20\) where \(x\) and \(y\) are measured in meters. Find the arc length of the cable between the two towers.

Finding the Volume of a Solid In Exercises\(37 - 40 ,\) find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(x\) -axis. Verify your results using the integration capabilities of a graphing utility. $$y = \sin x , \quad y = 0 , \quad x = 0 , \quad x = \pi$$

Centroid Explain why the centroid of a rectangle is the center of a rectangle.

Finding the Area of a surface of Revolution In Exercises \(45-48,\) write and evaluate the definite integral that represents the area of the surface generated by revolving the curve on the indicated interval about the \(y\) -axis. $$y=\sqrt[3]{x}+2, \quad 1 \leq x \leq 8$$

(a) Given a circular sector with radius \(L\) and central angle \(\theta\) (see figure), show that the area of the sector is given by \(S=\frac{1}{2} L^{2} \theta\) (b) By joining the straight-line edges of the sector in part (a), a right circular cone is formed (see figure) and the lateral surface area of the cone is the same as the area of the sector. Show that the area is \(S=\pi r L,\) where \(r\) is the radius of the base of the cone. (Hint: The arc length of the sector equals the circumference of the base of the cone.) (c)Use the result of part (b) to verify that the formula for the lateral surface area of the frustum of a cone with slant height \(L\) and radii \(r_{1}\) and \(r_{2}\left(\) see figure) is \(S=\pi\left(r_{1}+r_{2}\right) L\right.\) (Note: This formula was used to develop the integral for finding the surface area of a surface of revolution.)

Lateral Surface Area of a Cone A right circular cone is generated by revolving the region bounded by \(y=h x / r\) , \(y=h,\) and \(x=0\) about the \(y\) -axis. Verify that the lateral surface area of the cone is \(S=\pi r \sqrt{r^{2}+h^{2}}\)

Manufacturing A manufacturer drills a hole through the center of a metal sphere of radius \(R .\) The hole has a radius \(r\) Find the volume of the resulting ring.

Volume of a Sphere Use the disk method to verify that the volume of a sphere is \(\frac{4}{3} \pi r^{3},\) where \(r\) is the radius.

Volume of a Fuel Tank A tank on the wing of a jet aircraft is formed by revolving the region bounded by the graph of \(y=\frac{1}{8} x^{2} \sqrt{2-x}\) and the \(x\) -axis \((0 \leq x \leq 2)\) about the \(x\) -axis, where \(x\) and \(y\) are measured in meters. Use a graphing utility to graph the function. Find the volume of the tank analytically.