Areas Under Curves

This lesson covers the use of definite integration to find the area enclosed between a curve and the x-axis. This is one of the most important applications of integration at A-Level and is tested extensively on AQA papers.

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The Fundamental Idea

The definite integral of a function f(x) from a to b gives the signed area between the curve y = f(x), the x-axis, and the vertical lines x = a and x = b:

∫(from a to b) f(x) dx = F(b) − F(a)

where F(x) is an antiderivative (indefinite integral) of f(x).

If the curve lies above the x-axis on [a, b], the integral gives a positive value equal to the area.

If the curve lies below the x-axis on [a, b], the integral gives a negative value.

Key Point: The integral calculates signed area (also called net area). If the question asks for the total area (which it usually does), you must handle regions below the x-axis separately and take their absolute value.

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Example 1: Area Above the x-axis

Find the area under y = x² + 1 between x = 0 and x = 3.