Areas Between Curves

This lesson covers finding the area between two curves using definite integration. This extends the previous lesson's ideas and is a common question type on AQA A-Level papers.

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The Key Principle

If two curves y = f(x) and y = g(x) satisfy f(x) ≥ g(x) on the interval [a, b], then the area between them is:

Area = ∫(from a to b) [f(x) − g(x)] dx

This works because:

• ∫ₐᵇ f(x) dx gives the area under the upper curve.
• ∫ₐᵇ g(x) dx gives the area under the lower curve.
• Subtracting gives the area between them.

Key Point: Always subtract the lower curve from the upper curve. If you get this the wrong way round, the answer will be negative (just take the absolute value, but it is better practice to identify which curve is on top).

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Step-by-Step Method

1. Find the intersection points by solving f(x) = g(x). These give the limits of integration.
2. Determine which curve is on top in each interval (substitute a test value).
3. Set up and evaluate the integral ∫ₐᵇ [upper − lower] dx.

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