Calculus: Analytic Geometry and Calculus, with Vectors

4.2 Riemann sums and integrals 219

7 This problem requires us to attain a complete understanding of a more
complex situation. Let

f(x) =3 (2 <x<4)

f(x)=4 (4<x<5)

and let f(2), f(4), and f(5) be defined in any way that pleases (or displeases) the
fancy. Then Theorem 4.27 implies existence of the integral I in

I = 125 f (t) dt.

We want to find I, that is, to find the numerical value of I. Draw a figure show-
ing the interval 2 <= x 5 5 and, in addition, the graph off. Make a partition P of
the interval 2 =< x < 5 in which 4 is one of the partition points. Choose the
points tk in such a way that they are not at the ends of the intervals in which they
lie. Show that the terms in the Riemann sum RS can be split into two sums
RS1 and RS2 in such a way that RS1 contains those terms for which 2 < tk < 4
and RS2 contains those terms for which 4 < tk < 5. Show that RSl = 6 and
RS2 = 4 and hence that RS = 10. Our next step is to realize what we are
trying to do. We are not trying to prove that f is integrable and are not required
to prove that IRS - 11 is small whenever the norm of P is small. We are trying
to find I, and we can use the known fact that IRS - 11 must be near 0 whenever
the norm of P is near 0. Therefore IRS - 11 must be near 0 whenever P is a
partition of the type constructed above and the norm of P is near 0. But RS =
10 for each partition of the type constructed above, and it follows that I = 10.
Notice that we have, in the course of our work, proved that

125f(t) dt= J

"f(t) dt + f5f(t) dt.

Interpret the numerical results in terms of areas of rectangular regions.

8 Supposing that x1 < x2 < x3 and that k1 and k2 are constants, draw a

figure showing the interval x1 5 x 5 xs and a graph of the function f for which

f(xi) = f(x2) = f(xa) = 0 and

f(x) = kl (x1 < x < x2)

f(x)= k2 (x2 < x < X3)-

Show that

f-07

f(t) dt=f-xs f(t)dt + J1f(t) dt

= kl(x2 - xl) + k2(xa - x2)

Tell how the result can be interpreted in terms of areas of rectangular regions

when (a) k1 and k2 are both positive, (b) k1 > 0 and k2 < 0, and (c) k1 and k2

are both negative. Explain the manner in which these results can be extended

to step functions that have constant values over 3 or 300 intervals instead of just 2.

9 Tell why each of the following Riemann integrals exists or fails to exist.

(a) f of (2 + 3t + 4t2) dt

it+1

(b) fo t + 2dt

(c)f 11 tdt (d)lit dt

Hint: Use theorems given in the text.