Table of Contents
Finding the Area between Two Curves
The area between two curves can be found by integrating the functions that define the curves. The region between the curves is the area between the x-axis and the curves, minus the areas of the two individual curves.
Formula to Find the Area between Two Curves
The area between two curves can be found using the following formula:
A = ∫(f(x) − g(x))dx
Where:
A is the area between the curves
f(x) is the function for the first curve
g(x) is the function for the second curve
dx is the differential of x
Calculating Areas Between Two Curves by Integration
The area between two curves can be calculated by integration. The integration process involves finding the area of a set of small rectangles that cover the space between the curves. The height of each rectangle is equal to the difference in the y-values of the two curves at that point, and the width of each rectangle is equal to the difference in the x-values of the two curves at that point. Once the areas of all of the rectangles have been calculated, the total area between the curves can be found by adding up the individual rectangle areas.
Calculating Areas Between Curves Using Double Integrals
We can use double integrals to calculate the area between two curves. Recall that a double integral is defined as
where is a function of two variables and is a region in the coordinate plane.
In order to calculate the area between two curves, we need to find the intersection of the two curves. We can then use the double integral to calculate the area between the curves and the x-axis.
In the following example, we will calculate the area between the curves and the x-axis.
Example
Calculate the area between the curves and the x-axis.
To calculate the area between the curves and the x-axis, we need to find the intersection of the two curves. We can do this by solving the equation of the curves for .
We get the following intersection points:
(0, 2)
(3, 1)
(6, 0)
We can now use the double integral to calculate the area between the curves and the x-axis.
We get the following result:
Solved Example
Q) The average of two numbers is 10. The sum of the two numbers is 20. What is the value of the two numbers?
A) The two numbers are 5 and 15.
