Module 17  Applications of Integration 
Introduction  Lesson 1  Lesson 2  Lesson 3  SelfTest 
Lesson 17.2: Total Area and The Area Between Two Curves 
This lesson explores finding the total area under a curve and the area between two curves. The total area of a region that is both above and below the xaxis is found by separating the positive parts of the graph from the negative parts. Each part is integrated and the absolute values of the results are added together to find the total area. Examine the graph of y = sin x from 0 to 2 again. As shown in Lesson 17.1, , and . The total area bounded by y = sin x on the interval [0, 2 ] is 2 +  2  = 4 square units. Using Absolute Value to Find Total Area Another method used to find the total area is to integrate the absolute value of the function. You can find the absolute value, abs(), under
17.2.1 Evaluate
and interpret the result.
17.2.2 Find the total area bounded by y = x^{3} – 3x^{2} – x + 3 and the xaxis on the interval [0, 4]. Finding the Area between Two Curves Integrals can be used to find the area between two curves by evaluating the integral of the difference between the upper curve and the lower curve between their points of intersection. Find the area between y = 4 – x^{2} and y = x by following the procedure below.
The upper curve is y = 4 – x^{2} and the lower curve is y = x. Find the points where the curves intersect.
The area between the curves is given by
17.2.3 Evaluate the integral above on your TI89 and interpret the result. 
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