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Area between graphs

The area between two graphs is the integral of the upper one minus the lower one. The limits are the intersection points, so you first solve the equation between the functions. On the interval from 0 to 1 the straight line $y = x$ lies above the parabola $y = x^2$ . The integral of the difference becomes one half minus one third. A negative answer means that you have swapped which graph is on top.

Try it yourself

The integral as area

A = \int_a^b \bigl(f(x) - g(x)\bigr)\,dx

upper minus lower

Symbols

$f$	the upper graph
$g$	the lower graph
$a,\ b$	the intersection points

Example

$y = x$ and $y = x^2$ meet at $0$ and $1$ . $A = \int_0^1 (x - x^2)\,dx = \dfrac{1}{2} - \dfrac{1}{3} = \dfrac{1}{6} \approx 0.1667$ .

Find the intersection points before you integrate. They are the limits.

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Part of Mathematics R2: Area and volume with integrals.