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Home > CALC > Chapter 8 > Lesson 8.4.1 > Problem 8-142


The region M is enclosed by the functions and . Set-up, but do not evaluate, the integrals to represent the volumes of the solids formed by revolving M about each of the following axes.

  1. The -axis.

    Sketch the axis of rotation (flag pole) then use washers.

  1. The -axis.

    Use shells.

    Note the represents the length of the radius, it's part of the base equation. And represents the height of each shell.

  1. The line .

    Refer to the hint in part (a).

    Recall that R and r represent radii of an outer and inner solid (the 'inner' solid is the hole). Make R the function with the longer radii and r the function with the shorter radii. Note: It is possible that R is below r on the graph. Once rotation happens, this will switch.

    Also note that both radii have different lengths than and , since the axis of rotation is units above the -axis. Adjust the integral in part (a) accordingly:

  1. The line .

    As in part (b), to avoid using horizontal rectangles (and rewriting the integrand in terms of ), you can use shells.

    The setup should look exactly like the setup in part (b), with one exception: The radii are no longer the same values as the bounds. They will be either longer or shorter than the bounds. So shift the radii units to the left or right: