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Area Under A Curve




In this lessons, 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 rwo curves.

Related Topics: More Calculus Topics

Formula for Area 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

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,



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



Calculus - Finding the Area Under the Curve (1 of 7)
Find the area under the curve y = x2 from x = 0 to x = 4

Calculus - Finding the Area Under the Curve (2 of 7)
Find the area under the curve y = x3 from x = 1 to x = 3 and then from x = -2 to 2


 




Calculus - Finding the Area Under the Curve (3 of 7)
Find the area under the curve y = x2 - 2x + 8 from x = 1 to x = 2
Calculus - Finding the Area Under the Curve (4 of 7)
Find the area under the curve y = 1/x2 from x = 1 to x = 4
Calculus - Finding the Area Under the Curve (5 of 7)
Find the area under the curve y = sin x from x = 0 to x = π
Calculus - Finding the Area Under the Curve (6 of 7)
Find the area under the curve y = sin x cos x from x = π/4 to x = π/2
Calculus - Finding the Area Under the Curve (7 of 7)
Find the area under the curve y = x2/&rad;(x3 + 9) from x = -1 to x = 1


 



The Area Under a Curve: approx. the definite integral.
Approximate the area under the curve f(x) = x2 from x = 1 to 3. Use n = 4 rectangles.


Area under a curve using integration, step by step, example
Find the area bounded by the curves y = x2 - 6x + 9 and y = x + 3


Integration is used to determine the area under a curve.
A parabola, y = 2x - x2 is drawn such that it intersects the x-axis. The x-intercepts are determined so that the area can be calculated.


Introduction - Area under a curve - definite integral
Definite Integrals and 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.



Area between Two Curves

Finding Areas Between Curves


Area under a curve using integration, step by step, example.


Finding Areas Between Curves


Area Between Curves - Integrating with Respect to y



Use the following Area between Curves Calculator to show you the steps and to check your answers.




 



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