Chapter 13 Double and Triple Integrals 1004 exy dA, 18. where R = {(x, y) | 1 < x < 2 and 1 < y < 3} 13. State Fubini's theorem. 14. Explain why using an iterated integral to evaluate a dou- ble integral is often easier than using the definition of the double integral to evaluate the integral. / cos(xy) dA, R. 19. 15. Explain how the Fundamental Theorem of Calculus is used in evaluating the iterated integral S f(x, y) dy dx. {(+,y) 1 x드플 and 드y 드지 where R = 2 16. Explain how the Fundamental Theorem of Calculus is 20. Use the results of Exercises 18 and 19 to explain why 't 2 may not be possible to evaluate a double integral by us- used in evaluating the iterated integral " S"f(x, y) dx dy. 17. Earlier in this section, we showed that we could use Fubini's theorem to evaluate the integral lex²y dA and 21. Show that Sa Se f(x, y) dy dx does not always equal Sa Sa f(x, y) dx dy by evaluating these two iterated inte- we showed that x?y dx dy = 91. Now evaluate when a = c = 0 and b = d = 1 grals for f(x, y) = Why does this result not violate Fubini's theorem? 22. Outline the steps required to find the volhume solid bounded by the graph of a function f(x, v) an Noxy-plane for a < x
Chapter 13 Double and Triple Integrals 1004 exy dA, 18. where R = {(x, y) | 1 < x < 2 and 1 < y < 3} 13. State Fubini's theorem. 14. Explain why using an iterated integral to evaluate a dou- ble integral is often easier than using the definition of the double integral to evaluate the integral. / cos(xy) dA, R. 19. 15. Explain how the Fundamental Theorem of Calculus is used in evaluating the iterated integral S f(x, y) dy dx. {(+,y) 1 x드플 and 드y 드지 where R = 2 16. Explain how the Fundamental Theorem of Calculus is 20. Use the results of Exercises 18 and 19 to explain why 't 2 may not be possible to evaluate a double integral by us- used in evaluating the iterated integral " S"f(x, y) dx dy. 17. Earlier in this section, we showed that we could use Fubini's theorem to evaluate the integral lex²y dA and 21. Show that Sa Se f(x, y) dy dx does not always equal Sa Sa f(x, y) dx dy by evaluating these two iterated inte- we showed that x?y dx dy = 91. Now evaluate when a = c = 0 and b = d = 1 grals for f(x, y) = Why does this result not violate Fubini's theorem? 22. Outline the steps required to find the volhume solid bounded by the graph of a function f(x, v) an Noxy-plane for a < x
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
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Transcribed Image Text:Chapter 13
Double and Triple Integrals
1004
exy dA,
18.
where R = {(x, y) | 1 < x < 2 and 1 < y < 3}
13. State Fubini's theorem.
14. Explain why using an iterated integral to evaluate a dou-
ble integral is often easier than using the definition of the
double integral to evaluate the integral.
/ cos(xy) dA,
R.
19.
15. Explain how the Fundamental Theorem of Calculus is
used in evaluating the iterated integral S f(x, y) dy dx.
{(+,y) 1 x드플 and 드y 드지
where R =
2
16. Explain how the Fundamental Theorem of Calculus is 20. Use the results of Exercises 18 and 19 to explain why 't
2
may not be possible to evaluate a double integral by us-
used in evaluating the iterated integral " S"f(x, y) dx dy.
17. Earlier in this section, we showed that we could use
Fubini's theorem to evaluate the integral lex²y dA and
21. Show that Sa Se f(x, y) dy dx does not always equal
Sa Sa f(x, y) dx dy by evaluating these two iterated inte-
we showed that x?y dx dy = 91. Now evaluate
when a = c = 0 and b = d = 1
grals for f(x, y) =
Why does this result not violate Fubini's theorem?
22. Outline the steps required to find the volhume
solid bounded by the graph of a function f(x, v) an
Noxy-plane for a < x <b and c<y< d.
%3D
the double integral by evaluating the iterated integral
3 5
(x+y)3
SiSix²y dy dx.
Explain why it would be difficult to evaluate the double inte-
grals in Exercises 18 and 19 as iterated integrals.
Skills
Evaluate the sums in Exercises 23-28.
38.
y² dA,
3 2
3 2
where R = {(x, y) | –3 < x < 2 and -2 < y <2}
23. ΣΣ
ρ 24 ΣΣν
tarh olgmaxo bni j-1 k=1
Sl.e-
j=1 k=1
(2 – 3x2 + y²) dA,
R.
where R = {(x, y) |-3<x<2 and 3 < y < 5}
39.
3 4
25. ΣΣ6- 43)
26 ΣΣύ-bA
%3D
j=1 k=1
j=1 k=1
obivang
4 3 2
40.
(x - e) dA,
27, ΣΣΣ'':
28 ΣΣΣ:
where R = {(x, y)1-3<x<2 and -2sys2}
i=1 j31 k-1
i=1 j-1 k=1
si
sin(x+2y) dA,
41.
Jse Definition 13.4 to evaluate the double integrals in Exer-
R.
ises 29-32.
where R = {(x, y) | 0 < x < n and 0 <ys
xy dA
R.
where R = {(x, y) |0 <x< 2 and 1<y< 4}
9.
42.
x sin x cos y dA,
%3D
R.
where R = {(x, y)1-3<x< 2 and -2sys2
R.
43.
xe*y dA,
where R = {(x, y) |-1<x< 0 and 0 sys2}
R.
xy° dA
where R= {(x, y) |-25*S2 and -1sys 1}
Jl.
where R {(x, y)10sxs1 and 0sy s In5–
*
44.
x²
cos(xy) dA,
R.
where R= (x, u) 10 <x< n and 0syS
2.3
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