25. Above the cone z = √√x² + y² and below the sphere x² + y² + 2² = 1(x) 19bianco bau (Sowgi
25. Above the cone z = √√x² + y² and below the sphere x² + y² + 2² = 1(x) 19bianco bau (Sowgi
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
25
![12. cos
№p
origin and radius 2
13. arctan(y/x) dA, 1 sidshm
s√√x² + y² dA, where D is the disk with center the
2
where R = {(x, y) | 1 ≤ x² + y² ≤ 4, 0 ≤ y ≤ x}
14. xdA, where D is the region in the first quadrant that lies
4 and x² + y² = 2x
adiwollut oft bisulnys
—
15-18 Use a double integral to find the area of the region.
15. One loop of the rose r = cos 30
and r = 1 Cos 0
16. The region enclosed by both of the cardioids r = 1 + cos 0
to
2
circle x² + y² = 1
17. The region inside the circle (x - 1)² + y² = 1 and outside the
1
0
18. The region inside the cardioid r = 1 + cos 0 and outside the
18 circle r = 3 cos ad lliw air brus econtine to an
20. Below the cone z =
pogo
101
Sm
19.01
25ldensy mobiST OW] to anoitonut varansbill
19-27 Use polar coordinates to find the volume of the given solid.
x² + y² ≤ 25
19. Under the paraboloid z = x² + y² and above the disk
1 ≤ x² + y² ≤ 4
siduob Sul day boqiups
Moments
x² + y² and above the ring
108 vienob
21. Below the plane 2x + y + z = 4 and above the disk
ni (x² + y² ≤ 18
Jinu 19q 221
plane z = 7 in the first octant
Sing1536)
1536) 0 NOB
22. Inside the sphere x² + y² + z² = 16 and outside the
cylinder x² + y² = 4
2
MA
23. A sphere of radius a
AA
24. Bounded by the paraboloid z = 1 + 2x² + 2y² and the
4091
DIST
mil
25. Above the cone z = √x² + y² and below the sphere
x² + y² + z² = 1(x) 19biznos bas (Sugi
et AA
26. Bounded by the paraboloids z = 6 - x² - y² and
z = 2x² + 2y²6bbe sw 11. to
1225
27. Inside both the cylinder x² + y² = 4 and the ellipsoid
4x² + 4y² + z² = 64
33
mu
anouamezonge or
28. (a) A cylindrical drill with radius r₁ is used to bore a hole
through the center of a sphere of radius r2. Find the volume
of the ring-shaped solid that remains.
(b) Express the volume in part (a) in terms of the height h of
the ring. Notice that the volume depends only on h, not
-14 (629 23 mil
AA(
on ri or
SECTION 15.3 Double Integrals in Polar Coordinate
31. S² S² xy² dx dy
32. S² S0²
√2x-x2
√x² + y² dy dx
33-34 Express the double integral in terms of a sing
to four decimal places.
respect to r. Then use your calculator to evaluate the
33.
Det
(x² + y2)2
radius 1
dA, where D is the disk with center
34. xy √1 + x² + y² dA, where D is the portic
x² + y² ≤ 1 that lies in the first quadrant
10017
00
no fol
35. A swimming pool is circular with a 40-ft diam
is constant along east-west lines and increases
2 ft at the south end to 7 ft at the north end. Fi
water in the pool.
36. An agricultural sprinkler distributes water in a
of radius 100 ft. It supplies water to a depth o
at a distance of r feet from the sprinkler.
(a) If 0 < R≤ 100, what is the total amount
per hour to the region inside the circle of
at the sprinkler?
(b) Determine an expression for the average
per hour per square foot supplied to the r
circle of radius R.
37. Find the average value of the function f(x, y)
on the annular region a² < x² + y² ≤ b², w
38. Let D be the disk with center the origin and
the average distance from points in D to the
39. Use polar coordinates to combine the sum
(x
S
JUNE √√7 xy dy dx +
√2
Sv² for xy dy dx +
into one double integral. Then evaluate the
40. (a) We define the improper integral (over tl
ff
1 = e = (x² + y²) dA
R²
= 1²0 1²² e²^(²2² +1²)
18
=
affe
Da
lim
a-x
dy dx
²) dy
(x² + y²) dA
D
where Da is the disk with radius a and](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F34f16726-3fe1-4d31-969a-c834e2b15ccd%2F02f69105-5b70-4125-a4b2-3d7f2ec6fc19%2Fd4l5zci_processed.jpeg&w=3840&q=75)
Transcribed Image Text:12. cos
№p
origin and radius 2
13. arctan(y/x) dA, 1 sidshm
s√√x² + y² dA, where D is the disk with center the
2
where R = {(x, y) | 1 ≤ x² + y² ≤ 4, 0 ≤ y ≤ x}
14. xdA, where D is the region in the first quadrant that lies
4 and x² + y² = 2x
adiwollut oft bisulnys
—
15-18 Use a double integral to find the area of the region.
15. One loop of the rose r = cos 30
and r = 1 Cos 0
16. The region enclosed by both of the cardioids r = 1 + cos 0
to
2
circle x² + y² = 1
17. The region inside the circle (x - 1)² + y² = 1 and outside the
1
0
18. The region inside the cardioid r = 1 + cos 0 and outside the
18 circle r = 3 cos ad lliw air brus econtine to an
20. Below the cone z =
pogo
101
Sm
19.01
25ldensy mobiST OW] to anoitonut varansbill
19-27 Use polar coordinates to find the volume of the given solid.
x² + y² ≤ 25
19. Under the paraboloid z = x² + y² and above the disk
1 ≤ x² + y² ≤ 4
siduob Sul day boqiups
Moments
x² + y² and above the ring
108 vienob
21. Below the plane 2x + y + z = 4 and above the disk
ni (x² + y² ≤ 18
Jinu 19q 221
plane z = 7 in the first octant
Sing1536)
1536) 0 NOB
22. Inside the sphere x² + y² + z² = 16 and outside the
cylinder x² + y² = 4
2
MA
23. A sphere of radius a
AA
24. Bounded by the paraboloid z = 1 + 2x² + 2y² and the
4091
DIST
mil
25. Above the cone z = √x² + y² and below the sphere
x² + y² + z² = 1(x) 19biznos bas (Sugi
et AA
26. Bounded by the paraboloids z = 6 - x² - y² and
z = 2x² + 2y²6bbe sw 11. to
1225
27. Inside both the cylinder x² + y² = 4 and the ellipsoid
4x² + 4y² + z² = 64
33
mu
anouamezonge or
28. (a) A cylindrical drill with radius r₁ is used to bore a hole
through the center of a sphere of radius r2. Find the volume
of the ring-shaped solid that remains.
(b) Express the volume in part (a) in terms of the height h of
the ring. Notice that the volume depends only on h, not
-14 (629 23 mil
AA(
on ri or
SECTION 15.3 Double Integrals in Polar Coordinate
31. S² S² xy² dx dy
32. S² S0²
√2x-x2
√x² + y² dy dx
33-34 Express the double integral in terms of a sing
to four decimal places.
respect to r. Then use your calculator to evaluate the
33.
Det
(x² + y2)2
radius 1
dA, where D is the disk with center
34. xy √1 + x² + y² dA, where D is the portic
x² + y² ≤ 1 that lies in the first quadrant
10017
00
no fol
35. A swimming pool is circular with a 40-ft diam
is constant along east-west lines and increases
2 ft at the south end to 7 ft at the north end. Fi
water in the pool.
36. An agricultural sprinkler distributes water in a
of radius 100 ft. It supplies water to a depth o
at a distance of r feet from the sprinkler.
(a) If 0 < R≤ 100, what is the total amount
per hour to the region inside the circle of
at the sprinkler?
(b) Determine an expression for the average
per hour per square foot supplied to the r
circle of radius R.
37. Find the average value of the function f(x, y)
on the annular region a² < x² + y² ≤ b², w
38. Let D be the disk with center the origin and
the average distance from points in D to the
39. Use polar coordinates to combine the sum
(x
S
JUNE √√7 xy dy dx +
√2
Sv² for xy dy dx +
into one double integral. Then evaluate the
40. (a) We define the improper integral (over tl
ff
1 = e = (x² + y²) dA
R²
= 1²0 1²² e²^(²2² +1²)
18
=
affe
Da
lim
a-x
dy dx
²) dy
(x² + y²) dA
D
where Da is the disk with radius a and
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