Area Between Two Curves

Abstract

  • Determine where the functions cross paths
  • Take the integral
    • If needed, split the integral up into bits so that the top function is always on top
    • Remember to change the bounds of the definite integral as well
  • Sometimes, it is easier to split into horizontal rectangles, so you may have the right function - left function

We divide the interval into subintervals of equal length, . Observe that the area of one "slice" is .
Next, sum up the area of the small slices

This is the Riemann Sum of the function
Now we take the limit

Which is the definite integral of the function , so

Definition

The area between two curves is:

General Steps

  1. Determine which function is bigger on a certain interval. For example:

    • On ,
    • On ,

  2. Take the definite integral

    It's important that the upper function is subtracted from the lower function

Example

Find the area between the curve of and 

left=-1; right=2;
bottom=-1; top=2;
---
x^3
x^2

Intersections:
For all , , so

Example

Find the area of the region between and from .

left=-0.5; right=2;
bottom=-0.5; top=1.5;
---
y = \sin(x) \{0 \le x \le \frac{\pi}{2}\}
y = \cos(x) \{0 \le x \le \frac{\pi}{2}\}
x = 0 \{0 \le y \le 1\} | BLACK
x = \frac{\pi}{2} \{0 \le y \le 1\} | BLACK

To find which function is on top, we need to find the intersections.

(See Working with Sines and Cosines)

Example

Sometimes, it is easier to divide the region into horizontal rectangles, and integrate on

Find the area of the region between and on and 

left=-.5; right=4;
bottom=-.5; top=2;
---
y = \ln(x)
x = 3
(3, \ln(3)) | LABEL: (3, ln(3))

Left function: (see inverse)
Right function:

where we have applied one of the log laws from the "other" section

Example

Find the area between the curves and on 

left=-0.5; right=2.5;
bottom=-0.5; top=1.5;
---
y = x
y = \frac{2}{x} - 1
(1, 1) | LABEL

Right function:
Left function:

Intersection of the two curves:

So the intersection is at