Parametric Integration

This lesson covers how to find areas under curves defined by parametric equations. This extends your knowledge of integration to situations where the curve is not given as y = f(x) but in parametric form. The key idea is to replace dx with (dx/dt) dt and change the limits from x-values to t-values.

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The Parametric Integration Formula

If a curve is defined parametrically by x = f(t) and y = g(t), then the area under the curve between x = a and x = b is

Area = ∫ y dx = ∫ y (dx/dt) dt

The limits of integration must be converted from x-values to t-values.

Important Steps:

1. Write y and dx/dt in terms of t.
2. Find the t-values corresponding to the x-limits.
3. Integrate y × (dx/dt) with respect to t.
4. Ensure the area is positive — take the modulus if necessary, or reverse limits.

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Basic Examples

Example 1: Find the area under the curve x = t², y = 2t between x = 0 and x = 4.

dx/dt = 2t