Mathematics 150BL, Fall 2003

Assignment 2 -- Solids of Revolution

  1. When finding the volume of a solid of revolution of a graph like that of $f(x)=\sin x$ on $[0,\pi ]$ you can use the computer to graph the area being revolved about the $x-$axis as well as the solid for which you wish to compute the volume.

    1. Graph $f$ and adjust the viewing rectangle to show the graph only on the given interval.

    2. Graph the solid of revolution obtained when $f$ restricted to domain $[0,\pi ]$ is revolved about the $x-$axis. (Hint: To plot a solid of revolution generated by rotating $f(x)$ about the x-axis we create the vector MATH and plot it using Compute: Plot3D+Rectangular.)

  2. For the following, adjust the viewing rectangles to show only the graphs on the restricted domains.

    1. Graph $g(x)=\sqrt{x}$ on $[0,16]$.

    2. Graph the solid of revolution obtained when $g$ restricted to domain $[0,16]$ is revolved about the $x-$axis.

  3. For the following, adjust the viewing rectangles to show only the graphs on the restricted domains.

    1. Graph $h(x)=16-x^{2}$ on $[-4,4]$.

    2. Graph the solid of revolution obtained when $h$ restricted to domain $[-4,4]$ is revolved about the $x-$axis.

  4. You can think of solids of revolution as using a certain amount of surface area to enclose a certain amount of volume.

    This leads to the question of what function $f$ over what interval leads to a solid of revolution enclosing as much volume $V$ as possible while using as little surface area $S$ as possible. We can make this precise by studying the ratio MATH. It turns out that this fraction never goes above a certain limit regardless of the function $f$ that is used. We can use the computer to experiment with various possibilities for $f$ to see how much volume a solid of revolution can enclose using a certain amount of area.

    1. The assumption is made that $f(x)\geq 0$ on $a\leq x\leq b$, and that the graph of $f$ over this interval is being revolved about the $x-$axis to form a solid of revolution.

      Since the disk method applies here, the volume is: MATH. Evaluate the volumes of the solids in problems 1, 2, and 3 and record the results in the table below.

      MATH

    2. To find the total surface area for a given solid you must compute the "side" surface area given by MATH. Then add the areas of the disks, if any, that form the "ends" of the solids. The areas of these disks are $\pi (f(a))^{2}$ and $\pi (f(b))^{2}$. (Do you see why?) Thus, total surface area, which we denote here by MATH. Find the total surface areas for problems 1, 2, and 3 and record the results in the table above.

    3. Compute MATH for problems 1, 2, and 3 and record the results in the table above.

    4. Experiment with functions of your own that you think will maximize MATHand repeat parts (a) to (c) for those functions.

    5. Make a conjecture about the maximum possible value of MATH and what shape of curve gives this maximum.

This document created by Scientific Notebook 4.0.