Module 19  Applications of Integration  
Introduction  Lesson 1  Lesson 2  Lesson 3  SelfTest  
Lesson 19.2: Total Area and The Area Between Two Curves  
This lesson explores finding the total area bounded by a curve and the xaxis on a given interval. It also investigates finding 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 = sinx from 0 to 2 again. As shown in Lesson 19.1, , and . The total area bounded by y = sinx and the xaxis 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(), by pressing . Compare the graphs of y = sin(x) and y = sin(x) shown below in the [0,2 ] x [2, 2] window. Notice that the absolute has the effect of reflecting the part of the graph of sin(x) that is below the xaxis so that it is above the xaxis. The integral of sin(x) from 0 to 2 gives the total area bounded by the curve y = sin(x) and the xaxis between x = 0 and x = 2 .
Evaluate
and interpret the result.
The total area bounded by y = sinx and the xaxis on the interval [0, 2 ] is 4 square units. 19.2.1 Find the total area bounded by y = x^{3} – 3x^{2} – x + 3 and the xaxis on the interval [0, 4] by integrating the absolute value of y = x^{3}  3x^{2}  x + 3. Click here for the answer. Finding the Area between Two Curves Integrals can be used to find the area between two curves by evaluating the integral of the value of the upper curve minus 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.
19.2.2 What is the approximate xcoordinate of the right point of intersection? We now need to integrate the upper curve minus the lower curve. The approximate area between the curves is given by 19.2.3 Evaluate the integral above on your TI83 and interpret the result. Click here for the answer.


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