Calculus

Area Between Two Curves

Find the area enclosed by two curves using upper curve minus lower curve.

Main formula

Over an interval where \(f(x)\) is above \(g(x)\), the area between them is:

\[ A=\int_a^b\big(f(x)-g(x)\big)\,dx \]
y=f(x)y=g(x)R
Find the intersection points first; they usually define the limits.

Worked example

Find the area enclosed by \(y=4-x^2\) and \(y=x+2\).

  1. Intersections: \(4-x^2=x+2\Rightarrow x=-2,1\).
  2. On \([-2,1]\), \(4-x^2\) is above \(x+2\).
  3. \[A=\int_{-2}^{1}(2-x-x^2)\,dx=\left[2x-\frac{x^2}{2}-\frac{x^3}{3}\right]_{-2}^{1}=\frac{9}{2}.\]
Exam tip: if the curves cross more than twice, split the region into separate integrals.

Area between two curves

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Enclosed area worked example

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Area between curves: exam practice

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