Exercises

1. This exercise is intended to familiarize you with the procedures revolve, RevInt, LeftDisk and LeftInt as an approach to approximating the volume of a solid of revolution using the disk method. Apply the four commands as listed above to the function f defined by on the interval with n=10. Your answer should include the two numerical values and references to the two 3-dimensional plots.

2. The formula for the volume of a cone, namely, can be obtained by assigning the points , , and as the vertices of a right triangle and rotating that triangle about the x-axis.
   1. Following a similar approach as above, derive the formula for the volume of a frustrum of a cone. That is, a cone with the top cut off parallel to the base.
   2. Show that your formula can be expressed in the traditional handbook form: , where and are the areas of the two bases.

3. A torus - a solid shaped like a doughnut - is generated by revolving a circle about an axis that does not intersect the circle. By rotating the circle about the line y = 3, print a plot of the torus which depicts its doughnut-like image and compute its volume. Your answer should include the numerical value for the volume and a reference to the 3-dimensional plot.

4. A glass container is fashioned acording to the image obtained by rotating the graph of the function g defined by on the interval and x measured in centimeters.
   1. Obtain a plot of the the container in an upright position. Be sure to reference your plot.
   2. Determine the capacity of the container if filled to the brim.
   3. It is decided that Herky the minnow is to be allowed to use the container as a fish bowl. When Herky is placed in the container when it is filled to a depth of 4 cm, the depth increases to 4.5 cm. If the water weighs 1 gm/cm, how much in ounces does Herky weigh?
   In exercise 4, can also be expressed as .