1. Collect homework.
2. Notice that you do not have to concern yourself about where the curves go below the x-axis in finding the area between two curves. The negatives work themselves out. Recall that in determining the total area between a curve and the x-axis over a particular interval, you had to find out if the curve intersected the x-axis between the beginning and end of the interval. You do not need to do that here. Look at the example below:
Let the area of each region in the diagram be denoted by the capital letters, A, B, C, D, and E. To determine the shaded area between the two curves, you would integrate f(x) from x = a to x = b and then subtract the integral of g(x) over the same interval.
Note that B + C + D is the area we were seeking.
3. Example:
First, determine the points of intersection and draw a picture of the two relations. What do you notice about this drawing (that makes it different from yesterday's problem)?
If you try to integrate with respect to the x-axis, you will not find the total area. You must turn integrate with respect to the y-axis; in other words, the rectangles that you are summing are drawn horizontally.
Figure 1 shows the area that we are trying to find.
Figure 2 shows the area that we would compute if you evaluated the integral with respect to the x-axis.
Figure 3 shows that we are summing up rectangles which are drawn horizontally - not vertically.
Figure 4 shows the area "under" the parabola (the blue and yellow shading) and the area "under" the line (just the blue shaded region).
4. Here are the two formulas for finding areas between two curves:
Emphasize that you are summing areas of rectangles whose base is either dy or dx, and whose height is the difference between the two functions or relations.
5. Examples
