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Home > CALC > Chapter 8 > Lesson 8.3.1 > Problem 8-101


The graph at right is for . Write an integral that could compute the volume of the solid when this region is rotated about:

  1. The -axis.

    Vertical rectangles.

  2. The -axis.

    Horizontal rectangles would be very difficult (because the inverse of is NOT a function). So use the Shell Method.

  3. The line .

    By shells:

    Where did the come from?
    Recall that each cylindrical shell is a prism with dimensions: (base)(height)(width) .
    Well, represents the (radius) in the circumference part of the formula: .
    If we were rotating about the -axis (as in part (b)), then the radius would be the same as the bounds, .
    But since we are rotating about a line that is parallel to the -axis, we do not want the radius to be the same as the bounds.
    When we plug in the lower bound , we actually want a radius that is .
    And when we plug in the upper bound , we actually want a radius that is .
    will make this work!

  4. The line .

    Vertical rectangles with a hole in the center.