James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 5.2 The Definite Integral 387
y
y=ƒ
5.
y c
f sxd dx 1 y b
f sxd dx − y b
f sxd dx
a
c
a
0
a
c
b
x
This is not easy to prove in general, but for the case where f sxd > 0 and a , c , b
Property 5 can be seen from the geometric interpretation in Figure 15: the area under
y − f sxd from a to c plus the area from c to b is equal to the total area from a to b.
FIGURE 15
Example 7 If it is known that y 10
0 f sxd dx − 17 and y8 0
SOLUTION By Property 5, we have
y 8
f sxd dx 1 y 10
f sxd dx − y 10
f sxd dx
0
8
0
f sxd dx − 12, find y10
8
f sxd dx.
so y 10
f sxd dx − y 10
f sxd dx 2 y 8
f sxd dx − 17 2 12 − 5
8
0
0
n
Properties 1–5 are true whether a , b, a − b, or a . b. The following properties, in
which we compare sizes of functions and sizes of integrals, are true only if a < b.
Comparison Properties of the Integral
6. If f sxd > 0 for a < x < b, then y b
f sxd dx > 0.
a
7. If f sxd > tsxd for a < x < b, then y b
f sxd dx > y b
tsxd dx.
a
a
8. If m < f sxd < M for a < x < b, then
msb 2 ad < y b
f sxd dx < Msb 2 ad
a
y
M
m
0 a
FIGURE 16
y=ƒ
b
x
If f sxd > 0, then y b f sxd dx represents the area under the graph of f, so the geometric
a
interpretation of Property 6 is simply that areas are positive. (It also follows directly from
the definition because all the quantities involved are positive.) Property 7 says that a bigger
function has a bigger integral. It follows from Properties 6 and 4 because f 2 t > 0.
Property 8 is illustrated by Figure 16 for the case where f sxd > 0. If f is continuous,
we could take m and M to be the absolute minimum and maximum values of f on the
inter val fa, bg. In this case Property 8 says that the area under the graph of f is greater
than the area of the rectangle with height m and less than the area of the rectangle with
height M.
Proof of Property 8 Since m < f sxd < M, Property 7 gives
y b
m dx < y b
f sxd dx < y b
M dx
a
a
a
Using Property 1 to evaluate the integrals on the left and right sides, we obtain
msb 2 ad < y b
f sxd dx < Msb 2 ad
a
n
Property 8 is useful when all we want is a rough estimate of the size of an integral
with out going to the bother of using the Midpoint Rule.
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