Area Between Curves

Area is a quantity that expresses the extent of a two-dimensional surface or shape, or planar lamina, in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analog of the length of a curve (a one-dimensional concept) or the volume of a solid (a three-dimensional concept).

Area Between Curves

The area between a positive-valued curve and the horizontal axis, measured between two values $a$ and $b$ ($b$ is defined as the larger of the two values) on the horizontal axis, is given by the integral from $a$ to $b$ of the function that represents the curve. The area between the graphs of two functions is equal to the integral of one function, $f(x)$, minus the integral of the other function, $g(x)$: A=∫ba(f(x)−g(x))dxA = \int_a^{b} ( f(x) - g(x) ) \, dx where $f(x)$ is the curve with the greater y-value .

Area Between Two Graphs

The area between two graphs can be evaluated by calculating the difference between the integrals of the two functions.

Integration: Area Under a Curve

Integration can be thought of as measuring the area under a curve, defined by $f(x)$, between two points (here, $a$ and $b$).

Example

Find the area between the two curves $f(x)=x$ and $f(x)= 0.5 \cdot x^2$ over the interval from $x=0$ to $x=2$.

Two curves, $y=x$ and $y = 0.5 \cdot x^2$, meet at the points $(x_0,y_0)=(0,0) $ and $(x_1,y_1)=(2,2)$. Since $x > 0.5 \cdot x^2$ over the interval from $x=0$ to $x=2$, the area can be calculated as follows: 

$\displaystyle{A = \int_0^{2} ( x - \frac{1}{2} x^2 ) \, dx = \left [ \frac{1}{2} x^2- \frac{1}{6} x^3 \right ]_{x=0}^{x=2} = \frac{2}{3}}$