Area between two curves

Objective: Find the area between two curves

Method

Let's consider 2 functions f and g. We want to find the area limited by the curves of these functions and the verticals x = a and x = b.

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The area limited by the the curves of f and g  and the two verticals passing respectively by x = a and x = b can be found by subtracting the area under the curve of g from the area under the curve of g.

Let A be the area between the 2 curves we can write:
A = Area under f-Area under g
The area under f and limited by the verticals passing by a and b and the x-axis is given by

The area under g limited by the verticals passing by a and b and the x-axis is given by

Let's substitute area under f area under g in A

According to this result, we can make the following statement:

Let f and g be two continuous functions on the interval [a,b] and f(x)≥ g(x) for all values of x in this interval, then the area between the the curves of and g is given by

Example

Solution

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             
Practice 

Sketch the region enclosed by the curves of y = x²and y = x + 3. Find the area. 

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