May 21, 2021

Chapters

In this article, we will discuss how to calculate the area between a function and the x-axis.

## Area between a Function and the x-axis

There are two scenarios in which we can find the area between a function and the x-axis:

- When the function is non-negative, i.e. positive
- When the function is negative

Now, let us see what steps one should follow while calculating the area between the function and x-axis.

## Steps for Calculating Area Under the Curve

To compute the area under the curve f(x), one should follow the steps below:

Step 1 - Sketch the area

Step 2 - Find the boundaries a and b

Step 3 - Write the definite integral function

Step 4 - Integrate the function

In the next section, we will discuss how to calculate the area when the function is positive.

## When the Function is Positive

If the function f(x) is positive on an interval [a, b], then the area of the region bounded by the graph of the function f, the horizontal axis (x - axis) and the vertical lines x = a, and x = b is defined as follows:

Now, let us solve some examples in which we will calculate the area under the curve.

## Example 1

Calculate the area bounded by the function and the x-axis.

### Solution

Follow these steps to solve this example.

**Step 1 - Sketch the graph**

Sketch the graph. The graph of the function is given below:

**Step 2 - Find the boundaries**

To find the boundaries a and b of the function, we have to determine the x-intercepts of the curve. Although, from the graph above, we can clearly determine that the limits of integration are 3 and -3, however, we can check it again by equating the function equal to zero as shown below:

**Step 3 - Using the boundaries, write the definite integral function**

In this step, we will use the boundaries 3 and -3 and write the function in definite integral form like this:

**Step 4 - Integrate the function**

Write the function using the sum/difference property of definite integrals like this:

Now, to find the definite integral of the function, first, we will calculate the antiderivative of the function. The antiderivative of the function is . For ease of calculation, consider C = 0.

The fundamental theorem of calculus says that:

Substitute 3 and -3 in the antiderivative of the function:

Hence, the area bounded by the function and the x- axis is

## Example 2

Calculate the area of the region enclosed by the function xy = 36, the lines x = 6 and x = 12 and the x-axis.

### Solution

Follow these steps to compute the area.

**Step 1 - Sketch the graph**

The graph of the function is given below:

**Step 2 - Find the boundaries**

The boundaries x = 6 and x = 12 are already given in this problem.

**Step 3 - Using the boundaries, write the definite integral function**

In this step, we will use the boundaries 6 and 12 and write the function in definite integral form like this:

**Step 4 - Integrate the function**

Now, to find the definite integral of the function, first, we will compute the antiderivative of the function. The antiderivative of the function is . For ease of calculation, consider C = 0.

The fundamental theorem of calculus says that:

Substitute 6 and 12 in the antiderivative of the function like this:

## When the Function is Negative

If the function is negative in a closed interval [a, b] then the graph of the function is below the horizontal axis. The area of the function can be defined like this:

Now, let us solve some examples in which we are given a negative function.

## Example 1

Find the area of the function bounded by the graph of and x-axis.

### Solution

Follow these steps to solve this example.

**Step 1 - Sketch the graph**

Sketch the graph. The graph of the function is given below:

The downward parabola shows that the function is negative.

**Step 2 - Find the boundaries**

To find the boundaries a and b of the function, we have to determine the x-intercepts of the curve. Although, from the graph above, we can clearly determine that the limits of integration are 4 and 0, however, we can check it again by equating the function equal to zero as shown below:

or

**Step 3 - Using the boundaries, write the definite integral function**

In this step, we will use the boundaries 4 and 0 and write the function in definite integral form like this:

**Step 4 - Integrate the function**

Write the function using the sum/difference property of definite integrals like this:

Now, to find the definite integral of the function, first, we will calculate the antiderivative of the function. The antiderivative of the function is . For ease of calculation, consider C = 0.

The fundamental theorem of calculus says that:

Substitute 4 and 0 in the antiderivative of the function:

## Example 2

Calculate the area bounded by the curve y = cos x and the x-axis between and .

### Solution

Follow these steps to solve this example.

**Step 1 - Sketch the graph**

Sketch the graph. The graph of the function y = cos x is given below:

**Step 2 - Find the boundaries**

The boundaries and are already given in this problem.

**Step 3 - Using the boundaries, write the definite integral function**

In this step, we will use the boundaries and to write the function in definite integral form like this:

**Step 4 - Integrate the function**

Now, to find the definite integral of the function, first, we will compute the antiderivative of the function. The antiderivative of the function is . For ease of calculation, consider C = 0.

The fundamental theorem of calculus says that:

Substitute and in the antiderivative of the function like this: