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8.6 Integrals and Area

8.51   Theorem. Let be a piecewise monotonic function from an interval to . Then

Proof: We already know this result for monotonic functions, and from this the result follows easily for piecewise monotonic functions.

Remark Theorem 8.51 is in fact true for all integrable functions from to , but the proof is rather technical. Since we will never need the result for functions that are not piecewise monotonic, I will not bother to make an assumption out of it.

8.52   Theorem. Let and let be a piecewise monotonic function. Then the graph of is a zero-area set.

Proof: We will show that the theorem holds when is monotonic on . It then follows easily that the theorem holds when is piecewise monotonic on .

Suppose is increasing on . Let and let be the regular partition of into equal subintervals.

Then

and

Hence

Now , so it follows from the squeezing rule that the constant sequence converges to , and hence

Remark: Theorem 8.52 is actually valid for all integrable functions on .

8.53   Theorem (Area between graphs.) Let be piecewise monotonic functions on an interval such that for all . Let

Then

Proof: Let be a lower bound for , so that

Let

for all , and let

Then

It follows from translation invariance of area that

Let

Then , and

It follows from theorem 8.52 that and are almost disjoint, so

and thus

By theorem 8.51 we have

and

Thus

Remark: Theorem 8.53 is valid for all integrable functions and . This follows from our proof and the fact that theorems 8.51 and 8.52 are both valid for all integrable functions.

8.54   Example. We will find the area of the set in the figure, which is bounded by the graphs of and where

and

Now

Hence

It follows that the points and in the figure are

Also, since for all ,

(This is clear from the picture, assuming that the picture is accurate.) Thus

We have now found the area, but the answer is not in a very informative form. It is not clear whether the number we have found is positive. It would be reasonable to use a calculator to simplify the result, but my experience with calculators is that I am more likely to make an error entering this into my calculator than I am to make an error by doing the calculation myself, so I will continue. I notice that three terms in the answer are repeated twice, so I have

Thus the area is about From the sketch I expect the area to be a little bit smaller than , so the answer is plausible.

8.55   Exercise. The curve whose equation is
 (8.56)

is shown in the figure. Find the area enclosed by the curve.
(The set whose area we want to find is bounded by the graphs of the two functions. You can find the functions by considering equation (8.56) as a quadratic equation in and solving for as a function of .) A

8.57   Exercise. A Find the areas of the two sets shaded in the figures below:

8.58   Exercise. A Find the area of the shaded region.

Next: 9. Trigonometric Functions Up: 8. Integrable Functions Previous: 8.5 Change of Scale   Index
Ray Mayer 2007-09-07