OML Search

Area Under A Curve




 
In these lessons and solutions, we will learn how to use integrals (or integration) to find the areas under the curves defined by the graphs of functions. We also learn how to use integrals to find areas between the graphs of two functions.

We have also included calculators and tools that can help you calculate the area under a curve and area between two curves.

Related Topics: More Calculus Topics

Formula for Area bounded by curves (using definite integrals)

The Area A of the region bounded by the curves y = f(x), y = g(x) and the lines x = a, x = b, where f and g are continuous f(x) ≥ g(x) for all x in [a, b] is

formula for area bounded by curves

Example:

Find the area of the region bounded above by y = x2 + 1, bounded below by y = x, and bounded on the sides by x = 0 and x = 1.

Solution:

The upper boundary curve is y = x2 + 1 and the lower boundary curve is y = x.

Using the formula for area bounded by curves,



How to find the Area between Curves?

Example:

Find the area between the two curves y = x2 and y = 2xx2.

Solution:

Step 1: Find the points of intersection of the two parabolas by solving the equations simultaneously.

x2 = 2xx2
2x2 – 2x = 0
2x(x – 1) = 0
x = 0 or 1

The points of intersection are (0, 0) and (1, 1)

Step 2: Find the area between x = 0 and x = 1




How to use the Area Under a Curve to approximate the definite integral?
Example:
Approximate the area under the curve f(x) = x2 (i.e. the area between y = x2 and y = 0) from x = 1 to 3. Use n = 4 rectangles. How to find the area under a curve using integration, step by step, example
Example:
Find the area bounded by the curves y = x2 - 6x + 9 and y = x + 3. How to use integration to determine the area under a curve?
A parabola is drawn such that it intersects the x-axis. The x-intercepts are determined so that the area can be calculated.
Example:
Calculate the area enclosed by the curve y = 2x - x2 and the x-axis.


 
How to use Definite Integrals to find Area Under a Curve? Use the following Definite Integral Calculator to find the Area under a curve.
Enter the function, lower bound and upper bound.

How to Find Areas Between Curves?
Example:
Find the area bounded by the curves y = x2 - 4x and y = 2x. How to find area between curves by Integrating with Respect to y.
Example:
Find the area bounded by the curves x = y3 - y and x = 1 - y4. Use the following Area between Curves Calculator to show you the steps and to check your answers.




Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations.


You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.


OML Search


We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.


[?] Subscribe To This Site

XML RSS
follow us in feedly
Add to My Yahoo!
Add to My MSN
Subscribe with Bloglines