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

388 Chapter 5 Integrals
y
1
y=1
y=1/e
y=e– x2
Example 8 Use Property 8 to estimate y 1
e 2x 2 dx.
0
SOLUTION Because f sxd − e 2x 2 is a decreasing function on f0, 1g, its absolute maximum
value is M − f s0d − 1 and its absolute minimum value is m − f s1d − e 21 . Thus,
by Property 8,
e 21 s1 2 0d < y 1
e 2x 2 dx < 1s1 2 0d
or e 21 < y 1
e 2x 2 dx < 1
Since e 21 < 0.3679, we can write
0
0
0
1
x
0.367 < y 1
e 2x 2 dx < 1
0
n
FIGURE 17
The result of Example 8 is illustrated in Figure 17. The integral is greater than the area
of the lower rectangle and less than the area of the square.
1. Evaluate the Riemann sum for f sxd − x 2 1, 26 < x < 4,
with five subintervals, taking the sample points to be right endpoints.
Explain, with the aid of a diagram, what the Riemann
sum represents.
2. If
fsxd − cos x
0 < x < 3y4
evaluate the Riemann sum with n − 6, taking the sample
points to be left endpoints. (Give your answer correct to six
decimal places.) What does the Riemann sum represent?
Illustrate with a diagram.
3. If f sxd − x 2 2 4, 0 < x < 3, find the Riemann sum with
n − 6, taking the sample points to be midpoints. What does
the Riemann sum represent? Illustrate with a diagram.
4. (a) Find the Riemann sum for f sxd − 1yx, 1 < x < 2, with
four terms, taking the sample points to be right endpoints.
(Give your answer correct to six decimal places.) Explain
what the Riemann sum represents with the aid of a sketch.
(b) Repeat part (a) with midpoints as the sample points.
5. The graph of a function f is given. Estimate y 10
0
f sxd dx using
five subintervals with (a) right endpoints, (b) left endpoints,
and (c) midpoints.
y
1
6. The graph of t is shown. Estimate y 4 22
tsxd dx with six subintervals
using (a) right endpoints, (b) left endpoints, and
(c) midpoints.
y
1
1
7. A table of values of an increasing function f is shown. Use
the table to find lower and upper estimates for y 30
f sxd dx.
10
x 10 14 18 22 26 30
fsxd 212 26 22 1 3 8
8. The table gives the values of a function obtained from an
experiment. Use them to estimate y 9 3
f sxd dx using three
equal subintervals with (a) right endpoints, (b) left endpoints,
and (c) midpoints. If the function is known to be an
increasing function, can you say whether your estimates
are less than or greater than the exact value of the integral?
x
0
1
x
x 3 4 5 6 7 8 9
fsxd 23.4 22.1 20.6 0.3 0.9 1.4 1.8
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