Area between two curves

The area of the region enclosed between the graphs of two functions is obtained by integrating their difference (in absolute value): $A = \int_a^b|f(x)-g(x)|\,dx.$ If $f\ge g$ on the whole of $[a,b]$, the absolute value can be dropped: $A = \int_a^b(f-g)\,dx$. The endpoints $a,b$ are usually the abscissae of the intersection points of the two curves.

Example — Area between $f(x) = -x^2+4x$ and $g(x) = 1+x$

The intersections are found from $-x^2+4x = 1+x$, that is $x^2-3x+1=0$: $x_A\approx 0{,}38$ and $x_B\approx 2{,}62$. Over the interval the parabola lies above the line, so: $A = \int_{0{,}38}^{2{,}62}\bigl[(-x^2+4x)-(1+x)\bigr]\,dx = \int_{0{,}38}^{2{,}62}(-x^2+3x-1)\,dx.$

The green region between the parabola $f$ (blue) and the line $g$ (red): the area is the integral of the difference $f-g$ between the two intersections.

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Topics: Integrale
Concepts: Area tra due curve · Integrale definito
Skills: Geometria analitica · Integrare · Interpretare grafico