In calculus, an **area function **finds the area between a function and the x-axis. These are calculated with definite integrals. The area function is formulated differently depending on whether the area is above the x-axis (example 1), below the x-axis (example 2) or a combination of the two (example 3).

A couple of

**very useful online calculators**:

- This Desmos calculator will show you the different shadings for any function; You’ll want to make this your first step as it will show you whether you’re dealing with a positive area, negative area, or a combination.
- Desmos’s calculator will also calculate the definite integral for you. For steps, use Go to Symbolab’s Calculator, which I use in the examples below.

## Above The X-Axis

**Example question:** What is the area of the function f(x) = x^{2} between x = 1 and x = 5?

**Solution:**

Step 1: Graph the Area (using Desmos):

This confirms that we are dealing with a positive area, so we can use a straightforward integral:

Step 2: **Calculate the definite integral**. The Desmos calculator (Step 1) will give you a solution: 124/3 ≈ 41.333.

If you need the integration steps:

- Go to Symbolab’s Calculator.
- Click on the small grey box and type in 5 as an upper bound. Then click on the bottom box and type 1 as your lower bound (your “bounds of integration”.
- Type in your formula (for this example, that’s x
^{2}) between the integral symbol ∫ and “dx”. Then click the red “Go” button on the right.

## Area Function Example 2: Below The X-Axis

**Example question: **What is the area between the x-axis and the function f(x) = x^{3} between x = -1 and x = 0?

*Follow the exact same Steps in example 1.
However, remove the negative sign in front of the solution because the area must be positive *

**Solution:**

Step 1: Graph the Area (using Desmos):

This confirms that we are dealing with a negative area under the x-axis. In other to get a positive value we have to put a negative sign in front of the integral:

In other words, you’re taking the negative of the integral solution (a negative negative is a positive).

The solution is 0.25.

## Combination of Above *and* Below The X-Axis

If your graph has parts above and below the axis, like this graph of x^{3} for the interval [-1, 1]:

**Note the intervals where the graph is positive**. For this example, the graph above (x^{3}) is positive between [0, 1]. Calculate the area using the steps in example 1. You should get 0.25.**Note the intervals where the graph is negative.**For this example, the graph above (x^{3}) is negative between [-1, 0]. Calculate the area using the steps in example 2. You should get 0.25.**Add the two answers from (1) and (2) together**. 0.25 + 0.25 – 0.50.

*That’s it!*

**CITE THIS AS:**

**Stephanie Glen**. "Area Function: Definition, Examples" From

**CalculusHowTo.com**: Calculus for the rest of us! https://www.calculushowto.com/area-function-definition-examples/

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