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.

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.

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 .

Proof: Let be a lower bound for , so that

Let

for all , and let

It follows from translation invariance of area that

Let

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.

and

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.

is shown in the figure. Find the area enclosed by the curve.