Ellipse E is centered at the origin, and has a horizontal minor axis of length 4. If you rotate the portion of E which falls only in the first and second quadrants about the x-axis, the resulting rotational solid has volume . Find the length of E's major axis without using a calculator.

You may not use a graphing calculator • Difficulty

Solution

An ellipse centered at the origin with horizontal minor axes of length 4 and vertical major axis of length 2a has equation

Solve this equation for y in preparation for the Disc Method, which we'll use to find the rotational volume.

You should only include the positive radical when calculating the volume, since the problem restricts the bounded area to the first and second quadrants. The graph intercepts the x-axis at x = –2 and x = 2, but since the graph is y-symmetric, you can calculate the volume by doubling the value of the integral on the half-sized interval from x = 0 to x = 2.

This will be the volume of the rotated ellipse; you know that it should equal , so solve for a.

Remember, the major axis has length 2a, so the final answer is .