Area Between Curves

This lesson covers finding the area between two curves — a key application of definite integration required by the Edexcel A-Level Mathematics specification (9MA0). You need to identify which curve is on top, set up the correct integral, and handle regions that may need to be split.

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The Basic Principle

To find the area enclosed between two curves y = f(x) and y = g(x) from x = a to x = b, where f(x) ≥ g(x) throughout the interval:

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

In words: integrate the top curve minus the bottom curve.

This works because:

• ∫ from a to b of f(x) dx gives the area under f(x) down to the x-axis
• ∫ from a to b of g(x) dx gives the area under g(x) down to the x-axis
• Subtracting removes the unwanted region below g(x)

Key Point: You must always subtract the lower function from the upper function to get a positive area.

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Finding the Limits of Integration

The limits a and b are usually the x-coordinates where the two curves intersect. To find these: