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

524 Chapter 7 Techniques of Integration
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1. Let I − y 4 0
f sxd dx, where f is the function whose graph is
shown.
(a) Use the graph to find L 2, R 2, and M 2.
(b) Are these underestimates or overestimates of I?
(c) Use the graph to find T 2. How does it compare with I?
(d) For any value of n, list the numbers L n, R n, M n, T n, and I
in increasing order.
y
3
2
1
f
0 1 2 3 4
2. The left, right, Trapezoidal, and Midpoint Rule approximations
were used to estimate y 2 0
f sxd dx, where f is the
function whose graph is shown. The estimates were 0.7811,
0.8675, 0.8632, and 0.9540, and the same number of subintervals
were used in each case.
(a) Which rule produced which estimate?
(b) Between which two approximations does the true value
of y 2 0
f sxd dx lie?
y
1
0
y=ƒ
3. Estimate y 1 0 cossx 2 d dx using (a) the Trapezoidal Rule and
(b) the Midpoint Rule, each with n − 4. From a graph of the
integrand, decide whether your answers are underestimates
or overestimates. What can you conclude about the true
value of the integral?
4. Draw the graph of f sxd − sin( 1 2 x 2 ) in the viewing rectangle
f0, 1g by f0, 0.5g and let I − y 1 f sxd dx.
0
(a) Use the graph to decide whether L 2, R 2, M 2, and T 2
underestimate or overestimate I.
(b) For any value of n, list the numbers L n, R n, M n, T n, and
I in increasing order.
(c) Compute L 5, R 5, M 5, and T 5. From the graph, which do
you think gives the best estimate of I?
5–6 Use (a) the Midpoint Rule and (b) Simpson’s Rule to approximate
the given integral with the specified value of n. (Round your
2
x
x
answers to six decimal places.) Compare your results to the
actual value to determine the error in each approximation.
5. y 2 x
0 1 1 x dx, n − 10 6. y
x cos x dx, n − 4
2 0
7–18 Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and
(c) Simpson’s Rule to approximate the given integral with the
specified value of n. (Round your answers to six decimal places.)
7. y 2
sx 3 2 1 dx, n − 10 8. y 2 1
1 0 1 1 x dx, n − 8
6
9. y 2 e x
2
dx, n − 10
0 1 1 x
10. y y2
s 3 1 1 cos x dx, n − 4
0
11. y 4
x 3 sin x dx, n − 8 12. y 3
e 1yx dx, n − 8
0
13. y 4
sy cos y dy, n − 8 14. y 3 1
dt, n − 10
0 2 ln t
15. y 1
0
x 2
1 1 x dx, n − 10 16. y 3 sin t
dt, n − 4
4 1 t
17. y 4
lns1 1 e x d dx, n − 8
0
18. y 1
sx 1 x3 dx, n − 10
0
19. (a) Find the approximations T 8 and M 8 for the integral
y 1 0 cos sx 2 d dx.
(b) Estimate the errors in the approximations of part (a).
(c) How large do we have to choose n so that the
approximations T n and M n to the integral in part (a) are
accurate to within 0.0001?
20. (a) Find the approximations T 10 and M 10 for y 2
1 e 1yx dx.
(b) Estimate the errors in the approximations of part (a).
(c) How large do we have to choose n so that the approximations
T n and M n to the integral in part (a) are accurate
to within 0.0001?
21. (a) Find the approximations T 10, M 10, and S 10 for
y 0
sin x dx and the corresponding errors ET, EM, and ES.
(b) Compare the actual errors in part (a) with the error
esti mates given by (3) and (4).
(c) How large do we have to choose n so that the approximations
T n, M n, and S n to the integral in part (a) are
accurate to within 0.00001?
22. How large should n be to guarantee that the Simpson’s
Rule approximation to y 1 0 e x 2 dx is accurate to within
0.00001?
1
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