Area between Two Functions

The area between two functions is equal to the area of the function located above minus the area of the function that lies below. Mathematically, we can denote this area like this:

Example 1

Find the area of the space bounded by the parabola and the straight line that passes through the points A(−1, 0) and B(1, 4).

Solution

Step 1 - Find the equation of a straight line In this step, we will calculate the equation of a straight line passing through two points A and B. To do so, first, we should calculate the slope of the line that passes through the points A(-1, 0) and B(1, 4). To calculate the slope, we will use the following formula:

Substitute the values of the points A and B in the above formula:

Now, substitute this slope in the point intercept equation below:

Hence, the equation of the straight line is y = 2x + 2.

Step 3 - Calculate the boundaries The points at which the line intersect the parabolas will be the boundaries or limits of the function. As we can see from the above graph, that  the line intersects the parabola at

and

. Hence, these are limits of the function. Step 4 - Calculate the definite integral To calculate the definite integral, first, use the information from the previous steps to write the functions in the following form:

Rewrite the function

by using the sum/difference rule of definite integrals like this:

To compute  the definite integral, we will first find the antiderivative of the function. The antiderivative of the function is

Now, use the fundamental theorem of calculus:

Substitute 2 and 0 in the antiderivative of the function like this:

Calculate the area of the figure bounded by the function and the lines y = x, at x = 0 and x = 2.

Solution

Step 1 - Sketch the graph In this example, we are already given the equation of the line y = x. Hence, we don't need to calculate it. We will simply start by sketching the graph of the functions

, and

. You can see in the above graph that from x = 0 to x = 1, the straight line is above the parabola, and from x = 1 to x = 2, the straight line is below the parabola. Hence, we will compute the areas using these limits above and below the parabola separately.

Example 2

Find the area between two curves

and

.

Solution

Follow these steps to calculate the area. Step 1 - Sketch the graph The graph of the two curves is given below: Step 2 - Find the boundaries Calculate the boundaries of the function by equation both the equations like this:

or

Hence, the boundaries of the function are 0 and 2.