Substitutions in Multiple Integrals - Exercises,Multiple Integrals, Math Notes | EduRev

: Substitutions in Multiple Integrals - Exercises,Multiple Integrals, Math Notes | EduRev

 Page 1


15.7 Substitutions in Multiple Integrals 1135
EXERCISES 15.7
Finding Jacobians and Transformed Regions
for Two Variables
1. a. Solve the system
for x and y in terms of u and y. Then find the value of the
Jacobian 
b. Find the image under the transformation u = x - y,
0sx, yd>0su, yd.
u = x - y, y = 2x + y
of the triangular region with vertices (0, 0),
(1, 1), and in the xy-plane. Sketch the transformed
region in the uy-plane.
2. a. Solve the system
for x and y in terms of u and . Then find the value of the
Jacobian 0sx, yd>0su, yd.
y
u = x + 2y, y = x - y
s1, -2d
y = 2x + y
4100 AWL/Thomas_ch15p1067-1142  8/25/04  2:58 PM  Page 1135
Page 2


15.7 Substitutions in Multiple Integrals 1135
EXERCISES 15.7
Finding Jacobians and Transformed Regions
for Two Variables
1. a. Solve the system
for x and y in terms of u and y. Then find the value of the
Jacobian 
b. Find the image under the transformation u = x - y,
0sx, yd>0su, yd.
u = x - y, y = 2x + y
of the triangular region with vertices (0, 0),
(1, 1), and in the xy-plane. Sketch the transformed
region in the uy-plane.
2. a. Solve the system
for x and y in terms of u and . Then find the value of the
Jacobian 0sx, yd>0su, yd.
y
u = x + 2y, y = x - y
s1, -2d
y = 2x + y
4100 AWL/Thomas_ch15p1067-1142  8/25/04  2:58 PM  Page 1135
b. Find the image under the transformation 
of the triangular region in the xy-plane bounded
by the lines and Sketch the
transformed region in the uy-plane.
3. a. Solve the system
for x and y in terms of u and y. Then find the value of the
Jacobian 
b. Find the image under the transformation
of the triangular region in the xy-
plane bounded by the x-axis, the y-axis, and the line
Sketch the transformed region in the uy-plane.
4. a. Solve the system
for x and y in terms of u and y. Then find the value of the
Jacobian 
b. Find the image under the transformation 
of the parallelogram R in the xy-plane with
boundaries and Sketch
the transformed region in the uy-plane.
Applying Transformations to Evaluate
Double Integrals
5. Evaluate the integral
from Example 1 directly by integration with respect to x and y to
confirm that its value is 2.
6. Use the transformation in Exercise 1 to evaluate the integral
for the region R in the first quadrant bounded by the lines
and 
7. Use the transformation in Exercise 3 to evaluate the integral
for the region R in the first quadrant bounded by the lines
and 
8. Use the transformation and parallelogram R in Exercise 4 to eval-
uate the integral
6
R
2sx - yd dx dy.
-s1>4dx + 1.
y = y =-s3>2dx + 1, y =-s3>2dx + 3, y =-s1>4dx ,
6
R
s3x
2
+ 14xy + 8y
2
d dx dy
y = x + 1. y =-2x + 4, y =-2x + 7, y = x - 2,
6
R
s2x
2
- xy - y
2
d dx dy
L
4
0
 
L
x= s y>2d + 1
x= y>2
 
2x - y
2
 dx dy
y = x + 1. x =-3, x = 0, y = x ,
y =-x + y
u = 2x - 3y,
0sx, yd>0su, yd.
u = 2x - 3y, y =-x + y
x + y = 1.
u = 3x + 2y, y = x + 4y
0sx, yd>0su, yd.
u = 3x + 2y, y = x + 4y
x + 2y = 2. y = 0, y = x ,
y = x - y
u = x + 2y, 9. Let R be the region in the first quadrant of the xy-plane bounded
by the hyperbolas and the lines 
Use the transformation with and 
to rewrite
as an integral over an appropriate region G in the uy-plane. Then
evaluate the uy-integral over G.
10. a. Find the Jacobian of the transformation and
sketch the region in the uy-plane.
b. Then use Equation (1) to transform the integral
into an integral over G, and evaluate both integrals.
11. Polar moment of inertia of an elliptical plate A thin plate of
constant density covers the region bounded by the ellipse
in the xy-plane. Find the
first moment of the plate about the origin. (Hint: Use the transfor-
mation )
12. The area of an ellipse The area of the ellipse
can be found by integrating the function
over the region bounded by the ellipse in the xy-plane.
Evaluating the integral directly requires a trigonometric substitu-
tion. An easier way to evaluate the integral is to use the transfor-
mation and evaluate the transformed integral over
the disk in the uy-plane. Find the area this way.
13. Use the transformation in Exercise 2 to evaluate the integral
by first writing it as an integral over a region G in the uy-plane.
14. Use the transformation to evaluate the
integral
by first writing it as an integral over a region G in the uy-plane.
Finding Jacobian Determinants
15. Find the Jacobian for the transformation
a.
b.
16. Find the Jacobian of the transformation
a.
b.
17. Evaluate the appropriate determinant to show that the Jacobian of
the transformation from Cartesian to Cartesian xyz-
space is r
2
 sin f .
rfu-space
x = 2u - 1, y = 3y - 4, z = s1>2dsw - 4d.
x = u cos y, y = u sin y, z = w
0sx, y, zd>0su, y, wd
x = u sin y, y = u cos y.
x = u cos y, y = u sin y
0sx, yd>0su, yd
L
2
0
 
L
sy + 4d>2
y>2
 y
3
s2x - yde
s2x - yd
2
 dx dy
x = u + s1>2dy, y = y
L
2>3
0
 
L
2 - 2y
y
sx + 2yde
sy - xd
 dx dy
G: u
2
+ y
2
… 1
x = au, y = by
ƒsx, yd = 1
x
2
>a
2
+ y
2
>b
2
= 1
pab
x = ar cos u, y = br sin u .
x
2
>a
2
+ y
2
>b
2
= 1, a 7 0, b 7 0,
L
2
1
 
L
2
1
 
y
x
 dy dx
G: 1 … u … 2, 1 … uy … 2
x = u, y = uy,
6
R
 a
A
y
x
+2xyb dx dy
y 7 0 u 7 0 x = u>y, y = uy
y = x, y = 4x. xy = 1, xy = 9
1136 Chapter 15: Multiple Integrals
4100 AWL/Thomas_ch15p1067-1142  8/25/04  2:58 PM  Page 1136
Page 3


15.7 Substitutions in Multiple Integrals 1135
EXERCISES 15.7
Finding Jacobians and Transformed Regions
for Two Variables
1. a. Solve the system
for x and y in terms of u and y. Then find the value of the
Jacobian 
b. Find the image under the transformation u = x - y,
0sx, yd>0su, yd.
u = x - y, y = 2x + y
of the triangular region with vertices (0, 0),
(1, 1), and in the xy-plane. Sketch the transformed
region in the uy-plane.
2. a. Solve the system
for x and y in terms of u and . Then find the value of the
Jacobian 0sx, yd>0su, yd.
y
u = x + 2y, y = x - y
s1, -2d
y = 2x + y
4100 AWL/Thomas_ch15p1067-1142  8/25/04  2:58 PM  Page 1135
b. Find the image under the transformation 
of the triangular region in the xy-plane bounded
by the lines and Sketch the
transformed region in the uy-plane.
3. a. Solve the system
for x and y in terms of u and y. Then find the value of the
Jacobian 
b. Find the image under the transformation
of the triangular region in the xy-
plane bounded by the x-axis, the y-axis, and the line
Sketch the transformed region in the uy-plane.
4. a. Solve the system
for x and y in terms of u and y. Then find the value of the
Jacobian 
b. Find the image under the transformation 
of the parallelogram R in the xy-plane with
boundaries and Sketch
the transformed region in the uy-plane.
Applying Transformations to Evaluate
Double Integrals
5. Evaluate the integral
from Example 1 directly by integration with respect to x and y to
confirm that its value is 2.
6. Use the transformation in Exercise 1 to evaluate the integral
for the region R in the first quadrant bounded by the lines
and 
7. Use the transformation in Exercise 3 to evaluate the integral
for the region R in the first quadrant bounded by the lines
and 
8. Use the transformation and parallelogram R in Exercise 4 to eval-
uate the integral
6
R
2sx - yd dx dy.
-s1>4dx + 1.
y = y =-s3>2dx + 1, y =-s3>2dx + 3, y =-s1>4dx ,
6
R
s3x
2
+ 14xy + 8y
2
d dx dy
y = x + 1. y =-2x + 4, y =-2x + 7, y = x - 2,
6
R
s2x
2
- xy - y
2
d dx dy
L
4
0
 
L
x= s y>2d + 1
x= y>2
 
2x - y
2
 dx dy
y = x + 1. x =-3, x = 0, y = x ,
y =-x + y
u = 2x - 3y,
0sx, yd>0su, yd.
u = 2x - 3y, y =-x + y
x + y = 1.
u = 3x + 2y, y = x + 4y
0sx, yd>0su, yd.
u = 3x + 2y, y = x + 4y
x + 2y = 2. y = 0, y = x ,
y = x - y
u = x + 2y, 9. Let R be the region in the first quadrant of the xy-plane bounded
by the hyperbolas and the lines 
Use the transformation with and 
to rewrite
as an integral over an appropriate region G in the uy-plane. Then
evaluate the uy-integral over G.
10. a. Find the Jacobian of the transformation and
sketch the region in the uy-plane.
b. Then use Equation (1) to transform the integral
into an integral over G, and evaluate both integrals.
11. Polar moment of inertia of an elliptical plate A thin plate of
constant density covers the region bounded by the ellipse
in the xy-plane. Find the
first moment of the plate about the origin. (Hint: Use the transfor-
mation )
12. The area of an ellipse The area of the ellipse
can be found by integrating the function
over the region bounded by the ellipse in the xy-plane.
Evaluating the integral directly requires a trigonometric substitu-
tion. An easier way to evaluate the integral is to use the transfor-
mation and evaluate the transformed integral over
the disk in the uy-plane. Find the area this way.
13. Use the transformation in Exercise 2 to evaluate the integral
by first writing it as an integral over a region G in the uy-plane.
14. Use the transformation to evaluate the
integral
by first writing it as an integral over a region G in the uy-plane.
Finding Jacobian Determinants
15. Find the Jacobian for the transformation
a.
b.
16. Find the Jacobian of the transformation
a.
b.
17. Evaluate the appropriate determinant to show that the Jacobian of
the transformation from Cartesian to Cartesian xyz-
space is r
2
 sin f .
rfu-space
x = 2u - 1, y = 3y - 4, z = s1>2dsw - 4d.
x = u cos y, y = u sin y, z = w
0sx, y, zd>0su, y, wd
x = u sin y, y = u cos y.
x = u cos y, y = u sin y
0sx, yd>0su, yd
L
2
0
 
L
sy + 4d>2
y>2
 y
3
s2x - yde
s2x - yd
2
 dx dy
x = u + s1>2dy, y = y
L
2>3
0
 
L
2 - 2y
y
sx + 2yde
sy - xd
 dx dy
G: u
2
+ y
2
… 1
x = au, y = by
ƒsx, yd = 1
x
2
>a
2
+ y
2
>b
2
= 1
pab
x = ar cos u, y = br sin u .
x
2
>a
2
+ y
2
>b
2
= 1, a 7 0, b 7 0,
L
2
1
 
L
2
1
 
y
x
 dy dx
G: 1 … u … 2, 1 … uy … 2
x = u, y = uy,
6
R
 a
A
y
x
+2xyb dx dy
y 7 0 u 7 0 x = u>y, y = uy
y = x, y = 4x. xy = 1, xy = 9
1136 Chapter 15: Multiple Integrals
4100 AWL/Thomas_ch15p1067-1142  8/25/04  2:58 PM  Page 1136
18. Substitutions in single integrals How can substitutions in
single definite integrals be viewed as transformations of
regions? What is the Jacobian in such a case? Illustrate with an
example.
Applying Transformations to Evaluate
Triple Integrals
19. Evaluate the integral in Example 3 by integrating with respect to
x, y, and z.
20. V olume of an ellipsoid Find the volume of the ellipsoid
(Hint: Let and Then find the volume of
an appropriate region in uyw-space.)
21. Evaluate
over the solid ellipsoid
(Hint: Let and Then integrate over an
appropriate region in uyw-space.)
z = cw . x = au, y = by,
x
2
a
2
+
y
2
b
2
+
z
2
c
2
… 1.
9
ƒ xyz ƒ dx dy dz
z = cw . x = au, y = by,
x
2
a
2
+
y
2
b
2
+
z
2
c
2
= 1.
22. Let D be the region in xyz-space defined by the inequalities
Evaluate
by applying the transformation
and integrating over an appropriate region G in uyw-space.
23. Centroid of a solid semiellipsoid Assuming the result that
the centroid of a solid hemisphere lies on the axis of symmetry
three-eighths of the way from the base toward the top, show, by
transforming the appropriate integrals, that the center of mass
of a solid semiellipsoid 
lies on the z-axis three-eighths of the way from the base
toward the top. (You can do this without evaluating any of the
integrals.)
24. Cylindrical shells In Section 6.2, we learned how to find the
volume of a solid of revolution using the shell method; namely, if
the region between the curve and the x-axis from a to b
is revolved about the y-axis, the volume of the
resulting solid is Prove that finding volumes by
using triple integrals gives the same result. (Hint: Use cylindrical
coordinates with the roles of y and z changed.)
1
a
b
 2pxƒsxd dx .
s0 6 a 6 bd
y = ƒsxd
z Ú 0,
sx
2
>a
2
d + sy
2
>b
2
d + sz
2
>c
2
d … 1,
u = x, y = xy, w = 3z
9
D
sx
2
y + 3xyzd dx dy dz
1 … x … 2, 0 … xy … 2, 0 … z … 1.
1137
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15.7 Substitutions in Multiple Integrals
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