shell method in calculus

Question

Use the Shell Method to compute the volume of a solid obtained by rotating the region enclosed by the graphs of the functions y=x^2, y=32-x^2 and x=1 about the y-axis.

Answer 1

The point of intersection is at (4,16) in the 1st quadrant. The graphs are even, so the rotation will go through the same graph in the 2nd quadrant. The shells will be vertical with a thickness dx and a height equal to the difference in the curves: (32 - x²) - x² = 32 - 2x²

The volume of the added differential-thickness shells is obtained from the integral (from r1 to r2) of 2πhrdr. In this case the h is 32-2x² and rdr is xdx and the limits of integration are x = 1 to 4.

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