Page 291 - Calculus Workbook For Dummies

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Chapter 16: Ten Things to Remember about Integration If You Know What’s Good for You

Area Below the x-Axis Is Negative

If you want, say, the area below the x-axis and above y = sin x between π- and 0, the top
of a representative rectangle is on the x-axis (the function y = 0) and its bottom is on
sin x. Thus, the height of the rectangle is 0 – sin x, and you use the following definite
0 0
h
integral to get the area: #^ 0 - sinx dx, which equals, of course, - # sinx dx. So this
- π - π
negative integral gives you the ordinary positive area. And that’s why an ordinary posi-
tive integral gives you a negative area for the parts of a curve that are below the x-axis.

Integrate in Chunks

When you want the total area between two curves and the “top” function changes
because the curves cross each other, you have to use more than one definite integral.
Each place the curves cross defines the edge of an area you must integrate separately.
(If a function crosses the x-axis, you have to consider y = 0 as the second function and
the x-intercepts as the crossing points.)