Question

Suppose you revolve the plane region completely about the given line to sweep out a solid of revolution. Describe the solid and find its volume in terms of pi.

Answer 1

Dear HSM: I encourage you to start writing out for yourself and potential helpers what you already know. For example, this "plane region," a square, has a uniform height of 2 and width of 4. Which method of finding the volume would you be inclined to use for #39? You have at least three to choose from: shells, washers, disks. Which of these would you automatically eliminate up front, and why?
Note that #39 can be checked by using the formula for the volume of a right circular cylinder of radius r and height h. I suspect you must use calculus to solve this problem, but the formula just mentioned would serve as a convenient check.

Answer 2

im only in geometry so i dont know how to use calculus

Answer 3

OK: Try looking up the formula for the volume of a right circular cylinder.
In Problem #39, the diagram shows you the height of the cylinder, as well as the radius (or diameter, which is twice the radius).
What's the next step?

Answer 4

for a cylinder

\[V=\pi r^2h\]

Answer 5

if you revolve it around the x axis you get this:
you need to find which one is r and which is h on this drawing

Answer 6

radius =2
height =4

Answer 7

Right on. Good going! Can you now calculate the volume of this (horizontal) cylinder in terms of pi?

Answer 8

\[V=\pi 2^2(4)\]
\[V=\pi 4(4)\]
\[V=16\pi\]

Answer 9

Yes, HSM, that's the correct result for Problem #39. Take a look at the other problem. What are the relevant r (radius) and h (height) for that one?

Answer 10

ok um how do i rotate it

Answer 11

sorry for having to leave earlier.

to rotate it, you pretty much imagine it in your head.

if you revolve the 4x2 rectangle about the x-axis it will look like a cylinder.

adding picture for emphasis:
black rectangle is revolved around the axis . it will look like above.