Volumes of Revolution about the y-axis

In the previous lesson we rotated regions about the x-axis. Rotation about the y-axis is equally important in Further Mathematics and requires a slightly different approach. This lesson covers the disc/washer method adapted for y-axis rotation and introduces the shell method as an alternative.

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1 Disc Method about the y-axis

Derivation

When a curve x = g(y) (or equivalently y = f(x) rearranged) is rotated about the y-axis, horizontal slices at height y with thickness delta y form discs of radius x.

V = pi * integral from c to d of x^2 dy

where c and d are the y-limits of the region.

Key Step: You must express x in terms of y (i.e. x = g(y)) before using this formula. Rearranging y = f(x) to get x = f^(-1)(y) is often the first step.

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2 Worked Example 1 — Simple Rearrangement

Find the volume when the region bounded by y = x^2, the y-axis, and y = 4 is rotated about the y-axis.

Solution:

Rearrange: y = x^2 implies x^2 = y (taking x >= 0 since we are in the first quadrant).