Contents:

## What is an “Area Under the Curve?”

The **area under a curve** is the area between the line of a graph (which is often curved) and the x-axis.

In calculus, you find the area under the curve using definite integrals.

Watch the video for an overview of definite integrals:

## The Area Function

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 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!*

## How to Find the Area Under Curve in Excel

Watch the video or read the steps below:

Microsoft Excel doesn’t have functions to calculate definite integrals, but you can approximate this area by dividing the curve into smaller curves, each resembling a line segment. Use the following steps to calculate the area under a curve in Excel as the total area of the trapezoids under these line segments:

## How to Find the Area Under Curve in Microsoft Excel: Steps

Example question: Find the area under curve in Excel for the graph below, from x = 1 to x = 6.

Step 1: Choose a few data points on the** x-axis **under the curve (use a formula, if you have one) and list these values in Column A in sequence, starting from Row 1. In this example from the graph on the left, your x-values are 1, 2, 3, 4, 5 and 6. Ensure that the first and last data points chosen on the curve are its starting and ending points respectively.

Step 2: List the corresponding **y-axis** data points in Column B, aligning them row-wise with the values in Column A. For this example, I’m going to assume that you *don’t have the formula*. In other words, I’m guessing where the values lie based on the graph. I’ll use y = 1, 0.5, 0.33, 0.225, 0.2 and 0.19.

Step 3: Type the following** formula** into cell C1

**=(B1+B2)/2*(A2-A1)**

and copy this for all Column C cells till the second-last row of data. To copy, click cell C1 and then click and drag the little black box in the right hand corner. In this example, you have 6 data points so you would drag the formula to cell C5.

Step 4: Calculate the **sum of the totals** in Column C. In this example, click cell C6 and then click the summation sign “Σ” on the ribbon. The solution will appear in cell C6.

Step 5: Delete the last row in column c (not the total!—see the image below). The correct approximation will not show in the summation cell.

For this example, the solution is **1.85**.

*That’s it!*

## Tip

When finding the area under curve in Excel, keep the x-axis increments as small as possible. This improves the curve’s approximation and the accuracy of the area under the curve. In other words, *the more values you input into columns A and B, the more accurate your results will be*. By using trapezoids of equal width, i.e. equidistant data points on the x-axis, you can do away with the first column; the formula in Column C is simply C1=(B1+B2)/2. The total sum of the values in Column C can then be multiplied by this constant width to give the total area under the curve.

Check out our YouTube channel for more help!

## References

“Area under a curve” image created at Desmos.com.