Page 81 - Calc for the Clueless

Region IV

The two answers agree. The area is 15 1/3.

The last example for us is a region where the curves cross each other (the top curve becomes the bottom and
the bottom becomes the top). This is another reason you must draw the region.

Example 4—

Find the area of the region between y = x and y = x - 3x.
3
3
3
To find the limits, we set x - 3x = x. x - 4x = 0. x(x + 2) × (x- 2) = 0. So x =-2, 0, and 2. In region I, the top
3
curve is the cubic. So we get [(x - 3x) - (x)] dx from -2 to 0. In region II, we have the straight line as the top
curve. We get [x - (x - 3x)] dx from 0 to 2.
3

Region I

Region II

The total area is 4 + 4 = 8.