Practical examples

Often we want to calculate the area between two curves in the plane. We might, for example, want to calculate the total increase over time of one function compared to another. Or we might want to calculate the cross-sectional area of a complicated machine part. A couple of other examples are shown in the plots below.

On the left is a Lorenz curve. It's a way of keeping track of income or wealth distribution in a country. The blue area below the curve illustrates the ability of people who already have a certain amount of income (low on the left, high on the right) to earn more money. It shows that people with the least wealth have the lowest capacity to generate more income, and those with the most income, the highest capacity to generate more. In a "perfect" world, everyone would have the capacity to earn more in proportion to what they have, but in fact, in most countries, the more you have, the easier it is to accumulate more.

Calculating the area between curves

Calculating the area between two curves is pretty straightforward. The three panels below illustrate the process. On the left is a straightforward integral, which yields the yellow area under a curve of some smooth (actually differentiable) function, $f(x)$ between $x = a$ and $x = b$. The center panel shows the integral of another function, call it $g(x)$, within the same interval, yielding the blue area. The area between those two curves (dotted) is just the difference of the two integrals,

$$\int_a^b f(x) dx - \int_a^b g(x) dx = \int_a^b [f(x) - g(x)] dx$$

There are three issues you will have to contend with that will complicate things. They are

Example 1

Simple area, non-overlapping functions

Find the area between the two curves

$$\begin{align} f(x) &= x^2 + 3x + 1 \\[5pt] g(x) &= x^2 + 2x + 11 \end{align}$$

between $x = 1$ and $x = 3$.

We don't really need to sketch a graph of these functions to know that they don't intersect. They're both parabolas with the same positive leading coefficient, and the y-intercept of the first is 2 units higher than the second. So we set up the integral

$$A = \int_1^3 g(x) - f(x) \, dx$$

where

$$g(x) - f(x) = -x + 10$$

to give an easy integral:

$$A = \int_1^3 (-x + 10) \, dx$$

Now we evaluate that integral

$$ = -\frac{x^2}{2} + 10x \, \bigg|_1^3$$

Example 2

Area enclosed by two curves

Calculate the area enclosed by the functions

$$\begin{align} f(x) &= x^2 - 4x + 5 \\[5pt] g(x) &= -x^2 + 6x - 4 \end{align}$$

The graph looks like this:

In order to calculate the intersection points shown, we set the functions equal to each other and solve for x, which involves completing the square. (Don't forget completing the square!) Here's the whole procedure: First set the equation equal to zero, move the constant term to the right side and divide by the coefficient of x²:

$$\begin{align} 2x^2 - 10x + 9 &= 0 \\[5pt] 2x^2 - 10x &= -9 \\[5pt] x^2 - 5x &= \frac{-9}{2} \end{align} $$

Example 3

When curves cross in the interval of integration

Sometimes the curves cross in the interval of interest, yet we still want to calculate the area between the curves. This problem requires us to find all of the points of intersection and perform separate integrals for each region. Here's one such region, the region between

$$ \begin{align} f(x) &= sin(x) \\ \\ g(x) &= -\frac{2 \sqrt{2}}{3 \pi} + \frac{2 \sqrt{2}}{3} \end{align} $$

The graph of these curves looks like this.

Now the two curves, $f(x)$ and $g(x)$ intersect in three places, enclosing two regions. Notice the symmetry of this problem. The areas of the right and left enclose regions are the same, so we can (and always should) use that to our advantage. There's no need to calculate both areas. Just calculate one and multiply by 2.

The integral is set up like this:

$$A = \int_{\frac{\pi}{4}}^{\pi} [f(x) - g(x)] dx + \int_{\pi}^{\frac{5\pi}{4}} [g(x) - f(x)] dx$$

Practice problems