91Ó°ÊÓ

Find the areas of the surfaces generated by revolving the curves in Exercises \(13-23\) about the indicated axes. If you have a grapher, you may want to graph these curves to see what they look like. $$y=\sqrt{x}, \quad 3 / 4 \leq x \leq 15 / 4 ; \quad x \text { -axis }$$

The region between the curve \(y=2 / x\) and the \(x\) -axis from \(x=1\) to \(x=4\) is revolved about the \(x\) -axis to generate a solid. a. Find the volume of the solid. b. Find the center of mass of a thin plate covering the region if the plate's density at the point \((x, y)\) is \(\delta(x)=\sqrt{x}\) c. Sketch the plate and show the center of mass in your sketch.

Use the shell method to find the volumes of the solids generated by revolving the regions bounded by the curves and lines about the \(x\) -axis. $$y=\sqrt{x}, \quad y=0, \quad y=2-x$$

Use the shell method to find the volumes of the solids generated by revolving the regions bounded by the given curves about the given lines. \(y=3 x, \quad y=0, \quad x=2\) a. The \(y\) -axis b. The line \(x=4\) c. The line \(x=-1\) d. The \(x\) -axis e. The line \(y=7\) f. The line \(y=-2\)

On June 11, 2004, in a tennis match between Andy Roddick and Paradorn Srichaphan at the Stella Artois tournament in London, England, Roddick hit a serve measured at 153 mi / h. How much work was required by Andy to serve a 2 -oz tennis ball at that speed?

Is there a smooth (continuously differentiable) curve \(y=f(x)\) whose length over the interval \(0 \leq x \leq a\) is always \(\sqrt{2 a} ?\) Give reasons for your answer.

The region in the first quadrant that is bounded above by the curve \(y=1 / x^{1 / 4},\) on the left by the line \(x=1 / 16,\) and below by the line \(y=1\) is revolved about the \(x\) -axis to generate a solid. Find the volume of the solid by a. the washer method. b. the shell method.

A Bundt cake, well known for having a ringed shape, is formed by revolving around the \(y\) -axis the region bounded by the graph of \(y=\sin \left(x^{2}-1\right)\) and the \(x\) -axis over the interval \(1 \leq x \leq\) \(\sqrt{1+\pi} .\) Find the volume of the cake.

Derive the formula for the volume of a right circular cone of height \(h\) and radius \(r\) using an appropriate solid of revolution.

Derive the equation for the volume of a sphere of radius \(r\) using the shell method.