Calculus How To

Area Function: Definition, Examples

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Integral Calculus >

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).

area function

The area function can calculate an area (a) above the curve (b) below the curve or (c) both above and below.



A couple of very useful online calculators:

  1. 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.
  2. 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) = x2 between x = 1 and x = 5?

Solution:
Step 1: Graph the Area (using Desmos):
function with a positive area

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

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:

  1. Go to Symbolab’s Calculator.
  2. 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”.integral bounds
  3. Type in your formula (for this example, that’s x2) between the integral symbol ∫ and “dx”. Then click the red “Go” button on the right. symbolabs go

Area Function Example 2: Below The X-Axis

Example question: What is the area between the x-axis and the function f(x) = x3 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):


definite integral for negative area

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:
definite integral for negative area

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 x3 for the interval [-1, 1]:
above and below

  1. Note the intervals where the graph is positive. For this example, the graph above (x3) is positive between [0, 1]. Calculate the area using the steps in example 1. You should get 0.25.
  2. Note the intervals where the graph is negative. For this example, the graph above (x3) is negative between [-1, 0]. Calculate the area using the steps in example 2. You should get 0.25.
  3. 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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