Volumes of Revolution about the x-axis

When a region in the xy-plane is rotated through 2pi radians (360 degrees) about the x-axis, it sweeps out a three-dimensional solid of revolution. Integration allows us to compute the exact volume of such solids using the disc method.

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1 The Disc Method

Derivation

Consider the curve y = f(x) between x = a and x = b, with f(x) >= 0. Divide the interval [a, b] into n thin strips, each of width delta x.

Each strip, when rotated about the x-axis, forms a thin disc (cylinder) with:

• Radius = y = f(x)
• Thickness = delta x

The volume of one disc is approximately:

delta V = pi * y^2 * delta x = pi * [f(x)]^2 * delta x

Summing all discs and taking the limit as delta x -> 0 gives:

V = pi * integral from a to b of y^2 dx = pi * integral from a to b of [f(x)]^2 dx

Critical Warning: You must square y before integrating. A very common error is to compute pi * integral of y dx instead of pi * integral of y^2 dx.

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