AI - Area Between Two Curves Lesson

Area Between Two Curves

Finding the area of a region lying under the graph of a function may be extended to finding areas of regions lying between the graphs of two functions. The area between two curves is the planar region formed by the boundaries of two or more continuous curves over a given interval. Consider the region between two continuous curves y = f(x) and y = g(x) where f(x) > g(x) throughout an interval [a, b]. As before, the bounded region is approximated by partitioning the interval [a, b] into n subintervals each of width Δx. If each subinterval is approximated by a rectangle with base Δx and height f(x_i) - g(x_i), then the area of the ith rectangle is Δx [f(x_i) - g(x_i)]. The area of the region would then be the Riemann sum $\sum_{i=1}^{n}[f(x_{i}) -g(x_1)\Lambda x_{i}]$. Recall that this sum becomes a better and better approximation of the bounded area as the number of rectangles approaches infinity. This limiting area is $A=\lim_{x \to \infty } \sum_{i=1}^{n}[f(x_{i}) -g(x_1)\Lambda x_{i}]$.

More formally, if f and g are continuous on [a, b] and f(x) > g(x) for all x in [a, b], then the area of the region bound by the graphs of f and g and the vertical lines x = a and x = b is $A=\int_a^b\left[f\left(x\right)-g\left(x\right)\right]dx$. The following presentation provides examples of area problems based on both vertical and horizontal rectangular strips.

View the video on area between two curves

An economic application of the area between two curves is the Lorenz curve, a graphical representation of wealth distribution developed by American economist Max Lorenz in 1905. On the graph, a straight diagonal line represents perfect equality of wealth distribution. The Lorenz curve lies beneath it, showing the reality of wealth distribution. The difference between the straight line and the curved line is the amount of inequality of wealth distribution, a figure described by the Gini coefficient.

Areas Between Switching Curves

It is not always the case that curves bounding regions for which an area is sought remain in the same relative position to one another. To find the area between two curves y = f(x) and y = g(x) where f(x) > g(x) for some values of x and f(x) < g(x) for other values of x, the regions must be split into separate sections.

To find the area between two curves f(x) and g(x) where f(x) > g(x) for some values of x, but f(x) < g(x) for other values of x, split the region into separate regions and calculate the area of each region.

For example, $A+A_1+A_2+A_3+A_4$

Since, $f\left(x\right)-g\left(x\right)= \begin{cases} f\left(x\right)-g\left(x\right) \: when\: f(x)≥g(x) \\ g(x)-f(x)\:when\:g(x)≥f(x) \end{cases} \: then A=\int_a^b|f(x)-g(x)|dx$

Example

Find the area of the region bounded by the curves y = sin x and y = cos x, x = 0, and $x=\frac{\pi}{2}$.

Solution: $A=A_1+A^{_{_2}}\\ A=\int_0^{\frac{\pi}{2}}\left|\cos\left(x\right)-\sin\left(x\right)\right|dx$

These curves intersect with sin x = cos x, i.e. when $x=\frac{\pi}{4}$

$A=\int_0^\frac{\pi}{2}\left|\cos\left(x\right)-\sin\left(x\right)\right|dx\\ A=\int_0^\frac{\pi}{4}\left[\cos\left(x\right)-\sin\left(x\right)\right]dx+\int_\frac{\pi}{4}^\frac{\pi}{2}\left[\sin\left(x\right)-\cos\left(x\right)\right]dx\\ =\left(\sin x+\cos x\right)\Bigg]_0^\frac{\pi}{4}+\left(-\cos x-\sin x\right)\Bigg]_\frac{\pi}{4}^\frac{\pi}{2}\\ =\left(\frac{1}{\sqrt[]{2}}+\frac{1}{\sqrt[]{2}}-0+1\right)+\left(0-1+\frac{1}{\sqrt[]{2}}+\frac{1}{\sqrt[]{2}}\right)=2\sqrt[]{2}-2$

Note-Since this region is symmetric about $x=\frac{\pi}{4}$, then $A=2A_1$

The following video reviews properties of definite integrals, area bounded by two graphs including switching curves, and the fnInt function on the TI-84.

Area Between Two Curves Practice

Area Between Two Curves: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of areas between two curves.

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