91Ó°ÊÓ

A solid is generated by revolving the region bounded by \(y=\sqrt{9-x^{2}}\) and \(y=0\) about the \(y\) -axis. A hole, centered along the axis of revolution, is drilled through this solid so that one-third of the volume is removed. Find the diameter of the hole.

A torus is formed by revolving the region bounded by the circle \(x^{2}+y^{2}=1\) about the line \(x=2\) (see figure). Find the volume of this "doughnut-shaped" solid. (Hint: The integral \(\int_{-1}^{1} \sqrt{1-x^{2}} d x\) represents the area of a semicircle.)

Use the Theorem of Pappus to find the volume of the solid of revolution. The torus formed by revolving the circle \((x-5)^{2}+y^{2}=16\) about the \(y\) -axis

A cone of height \(H\) with a base of radius \(r\) is cut by a plane parallel to and \(h\) units above the base. Find the volume of the solid (frustum of a cone) below the plane.

A sphere of radius \(r\) is generated by revolving the graph of \(y=\sqrt{r^{2}-x^{2}}\) about the \(x\) -axis. Verify that the surface area of the sphere is \(4 \pi r^{2}\).

The region bounded by \(y=\sqrt{x}, y=0, x=0\), and \(x=4\) is revolved about the \(x\) -axis. (a) Find the value of \(x\) in the interval \([0,4]\) that divides the solid into two parts of equal volume. (b) Find the values of \(x\) in the interval \([0,4]\) that divide the solid into three parts of equal volume.

What is a planar lamina? Describe what is meant by the center of mass \((\bar{x}, \bar{y})\) of a planar lamina.

State the Theorem of Pappus.

Water Depth in a Tank A tank on a water tower is a sphere of radius 50 feet. Determine the depths of the water when the tank is filled to one-fourth and three-fourths of its total capacity. (Note: Use the zero or root feature of a graphing utility after evaluating the definite integral.)

Individual Project \(\quad\) Select a solid of revolution from everyday life. Measure the radius of the solid at a minimum of seven points along its axis. Use the data to approximate the volume of the solid and the surface area of the lateral sides of the solid.