Exercises

1. For the function $$f(x) = \frac{\exp(-x/10)}{2+\sin(2x)}$$ over the interval $-1 \leq x \leq 6$,

   A

   Plot $f(x)$ over the given interval.

   B

   Plot the approximation of the solid of revolution using LeftDisk with 12 disks.

   C

   Plot the solid formed by revolving $f(x)$ about the $x$-axis.

   D

   Find the exact volume of the solid of revolution using the RevInt command and label your output exact.

   E

   Find the number of subintervals needed to approximate the volume of the solid of revolution about the $x$-axis using LeftInt with error no greater than 0.1.

2. Prove each of the volume formulas below by finding a function such that when revolved about the x-axis creates the solid whose volume you are trying to prove. Use the formula for the Disk method $$\pi \int_{a}^{b} (f(x))^2 \, dx$$ instead of the RevInt command.

   A

   The volume of a sphere of radius $r$ is $$\frac{4}{3} \pi r^3$$

   B

   The volume of a cylinder of radius $r$ and height $h$ is $\pi r^2h$.

   C

   The volume of a right circular cone of radius $r$ and height $h$ is $$\frac{\pi r^2h}{3}$$

3. A maker of brass candlesticks plans to make a candlestick whose shape is given by revolving the function
   
   $$f(x) = \frac{2}{x^2}+\frac{1}{5-x}+\frac{1}{4}\sin(\pi x)$$
   
   
   about the $x$ axis over the interval $0.9 \leq x \leq 4.4$. Plot the candlestick and find its volume.