บทที่ 1
Partial Derivatives
1.1
Function of Two or More Variables
y
1. Let f (x, y, z) = xy 2 z 3 + 3. Find f (x + y, x − y, x2 ) and f (xy, , xz).
x
2. Find the natural domain of f .
1
,
3x − y
1
y
(b) f (x, y) = +
,
x x−6
(a) f (x, y) =
(c) f (x, y) = x2 + y 2 +
x2
1
,
+ y2
(d) f (x, y) = ln xy,
(e) f (x, y) = ln x + ln y,
(f) f (x, y) = ln(1 − x2 − y 2 ),
√
(g) f (x, y) = xe− y−2 ,
√
(h) f (x, y) = x arcsin y,
2
.
(i) f (x, y) = √
x−y−5
3. Sketch the graph of f .
(a) f (x, y) = sin x,
(b) f (x, y) = −2x − 2y + 4,
(c) f (x, y) = x2 + y 2 ,
(d) f (x, y) = 4 − x2 − y 2 ,
p
(e) f (x, y) = − 1 − x2 − y 2 ,
(f) f (x, y) = 4x2 + y.
1
1.2
Limit and Continuity
4. Show that the limit does not exist.
xy
,
+ 2y 2
x−y
,
(b)
lim
(x,y)→(0,0) x2 + y 2
(a)
(c)
lim
(x,y)→(0,0) 3x2
x2 − y 2
,
(x,y)→(0,0) x2 + y 2
lim
xy 2
,
(x,y)→(0,0) x2 + y 2
3
.
(e)
lim
2
(x,y)→(0,0) x + 2y 2
(d)
lim
5. Evaluate the limit.
(a)
(b)
lim
cos(xy),
(x,y)→(0,0)
lim
2x4 y 3 − 5x3 y + 3xy 2 + x,
(x,y)→(1,3)
xy 3
,
(x,y)→(2,1) x + y
x−y
(d)
lim √
√ ,
(x,y)→(0,0) x + y
(c)
lim
(e)
x4 − 16y 4
,
(x,y)→(0,0) x2 + 4y 2
(f)
x2 − 2xy + y 2
,
x−y
(x,y)→(1,−1)
(g)
x2 − 2xy + y 2
,
x−y
(x,y)→(0,0)
(h)
1 − cos(x2 + y 2 )
,
x2 + y 2
(x,y)→(0,0)
(i)
lim
lim
lim
lim
lim
(x2 + y 2 ) ln(x2 + y 2 ).
(x,y)→(0,0)
6. Determine whether f is continuous at (x0 , y0 ).
(a) f (x, y) = x2 y 3 + 4x3 cos y + x2 y sin x, (x0 , y0 ) = ( π2 , π),
xy 3
, (x0 , y0 ) = (3, 2),
x+y
p
(c) f (x, y) = sin x2 + 5y 2 , (x0 , y0 ) = (1, −1),
(b) f (x, y) =
(d) f (x, y) =
ln(x2 + y 2 )
, (x0 , y0 ) = (0, −2),
2x + 8y
2

x2 + 7y 2 , (x, y) 6= (0, 0)
(x0 , y0 ) = (0, 0),
(e) f (x, y) =
−4
, (x, y) = (0, 0)

x−y

√
√ + 5 , (x, y) 6= (0, 0)
x+ y
(f) f (x, y) =
(x0 , y0 ) = (0, 0),

5
, (x, y) = (0, 0)
 2
2

 x − 2xy + y
, (x, y) 6= (0, 0)
x−y
(g) f (x, y) =
(x0 , y0 ) = (0, 0).

0
, (x, y) = (0, 0)
7. Describe the largest region D on which the function f is continuous.
(a) f (x, y) = y ln(1 + x),
√
(b) f (x, y) = x − y,
(c) f (x, y) = arcsin(xy),
(d) f (x, y) = e1−xy .
1.3
8. Find
Partial Derivatives
∂f
∂f
(x0 , y0 ) and
(x0 , y0 ) of the following functions.
∂x
∂y
(a) f (x, y) = 2x5 y 3 − 7x2 y 2 + xy + 5x, (x0 , y0 ) = (1, 2),
(b) f (x, y) = sin(xy), (x0 , y0 ) = (π, π),
π π
(c) f (x, y) = tan(x + y), (x0 , y0 ) = ( , ),
2 2
2y
2x
(d) f (x, y) = e
+ xy, (x0 , y0 ) = (1, 1),
3xy + 1
(e) f (x, y) =
, (x0 , y0 ) = (2, 3),
x+y
p
(f) f (x, y) = x2 + y 2 + 16, (x0 , y0 ) = (0, 3).
9. Find
∂z
∂z
and
of the following functions.
∂x
∂y
(a) z = 10x2 y 4 − 6xy 2 + 10x2 ,
(b) z = (x2 + 5x − 2y)8 ,
1
(c) z = 2
,
xy − x2 y
√
(d) z = xe
15xy ,
(e) z = sin(5x3 y + 7xy 2 ).
10. จงหาความชันของพื้นผิว z = f (x, y) ในทิศทางของ x และ y ที่จุด (x0 , y0 ) ของฟังก์ชัน f ดังต่อไปนี้
3
(a) f (x, y) = xe−y + 5y, (x0 , y0 ) = (3, 0),
(b) f (x, y) = sin(y 4 − 4x), (x0 , y0 ) = (2, 1),
(c) f (x, y) = (x + y)−1 , (x0 , y0 ) = (−2, 4).
11. By differentiating implicitly, find
∂z
∂z
(x0 , y0 ) and
(x0 , y0 ) of the followinf functions.
∂x
∂y
(a) x2 + y 2 − z 2 = 16, (x0 , y0 ) = (0, 5),
p
(b) z = 29 − x2 − y 2 , (x0 , y0 ) = (4, 3),
(c) z 3 + 4x + y 2 = 0, (x0 , y0 ) = (1, 2).
12. By differentiating implicitly, find
13. Find
∂z
of the function z 3 + 2z 2 + yz + x2 y = 5 at (x0 , y0 , z0 ) = (1, 1, 1).
∂x
∂f ∂f
∂f
,
and
of the following functions.
∂x ∂y
∂z
(a) f (x, y, z) = x2 y 4 z 3 + xy + z 2 + 1,
(b) f (x, y, z) = z ln(x2 y cos z),
(c) f (x, y, z) = yez sin(xz),
x2 − y 2
.
(d) f (x, y, z) = 2
y + z2
∂f
for all i = 1, 2, . . . , n of the following functions.
∂xi
√
(a) f (x1 , x2 , . . . , xn ) = x1 + x2 + . . . + xn ,
14. Find
(b) f (x1 , x2 , . . . , xn ) = (x1 x2 . . . xn )2 .
15. Find the mixed second-order partial derivatives of the following functions.
(a) f (x, y) = 4x2 − 2y + 7x4 y 5 ,
(b) f (x, y) = sin(3x2 + 6y 2 ),
(c) f (x, y) = xe2y .
16. Express thr following derivatives in ∂ notation.
(a) fxxyy
(b) fyxyy
(c) fxyxy
17. Find the indicated partial derivatives.
(a) Find fyxy and fxxy of the function f (x, y) = x3 y 5 − 2x2 y + x,
∂3f
∂3f
(b) Find
and
of the function f (x, y) = ey cos x,
∂x2 ∂y
∂y 2 ∂x
∂4f
(c) Find fxyy and
of the function f (x, y) = y 3 e−5x .
∂x2 ∂y 2
4
1.4
Chain Rule
18. Use an appropriate form of the chain rule to find
dz
.
dt
1
(a) z = e1−xy ; x = t 3 , y = t3 ,
1
(b) z = 3 cos x − sin(xy); x = , y = 3t,
t
p
4
(c) z = 1 + x − 2xy ; x = ln t, y = t,
√
2
(d) z = ln(2x2 + y); x = t, y = t 3 .
19. Use an appropriate form of the chain rule to find
1
dw
.
dt
2
(a) w = ln(3x2 − 2y + 4z 3 ); x = t 2 , y = t 3 , z = t−2 ,
(b) w = x sin(yz 2 ); x = cos t, y = t2 , z = et ,
p
(c) w = 1 + x − 2yz 4 x; x = ln t, y = t, z = 4t,
(d) w = 5 cos(xy) − sin(xz); x = 1t , y = t, z = t3 .
20. Use an appropriate form of the chain rule to find
∂z
∂z
and
.
∂u
∂v
(a) z = 8x2 y − 2x + 3y; x = uv, y = u − v,
(b) z = x2 − y tan x; x = uv , y = u2 v 2 ,
(c) z = 3x − 2y; x = u + v ln u, y = u2 − v ln v,
√
1
2
(d) z = ex y ; x = uv, y = ,
v
(e) z = cos x sin y; x = u − v, y = u2 + v 2 ,
x
(f) z = ; x = 2 cos u, y = 3 sin v.
y
21. Use a chain rule to find
and z = 2u − v.
∂w
∂w
and
of the function w = 3xy+2 cos(yz)+ex+y+z where x = uv, y = u+v
∂u
∂v
∂w ∂w
∂w
,
and
of the function w = 4x2 + 4y 2 + z 2 where x = ρ sin φ cos θ,
∂ρ ∂φ
∂θ
y = ρ sin φ sin θ and z = ρ cos φ.
x
∂z ∂z 23. Use a chain rule to find
and
of the function z = xye y where x = r cos θ and
π
π
∂r r=2, θ= 6
∂θ r=2, θ= 6
y = r sin θ.
22. Use a chain rule to find
√
dw
of the function w = 3xy 2 z 3 where y = 3x2 + 2 and z = x − 1.
dx
√
dw 25. Use a chain rule to find
1 of the function w = r2 − r tan θ where r = s and θ = πs.
ds s= 4
∂f ∂f 26. Use a chain rule to find
and
of the function f (x, y) = x2 y 2 − x + 2y where
∂u
∂v
u=1,
v=−2
u=1,
v=−2
√
x = u and y = uv 3 .
24. Use a chain rule to find
5
1.5
Differentiability, Differentials and Applications
27. Show that the following functions are differentiable at (2, 1).
(a) f (x, y) = 2x2 y + 3,
(b) f (x, y, z) = xyz.
28. Show that the following functions are differentiable everywhere.
(a) f (x, y) = x2 sin y,
(b) f (x, y, z) = xy cos z.
29. Use a total differential to approximate the change in the values of f from P to Q.
(a) f (x, y) = x2 + 2xy − 4x ; P = (1, 2), Q = (1.01, 2.04),
√
(b) f (x, y) = ln 1 + xy ; P = (0, 2), Q = (−0.09, 1.98),
1
1
(c) f (x, y) = x 3 y 2 ; P = (8, 9), Q = (7.78, 9.03),
x+y
; P = (−1, −2), Q = (−1.02, −2.04),
(d) f (x, y) =
xy
(e) f (x, y, z) = 2xy 2 z 3 ; P = (1, −1, 2), Q = (0.99, −1.02, 2.02),
xyz
(f) f (x, y, z) =
; P = (−1, −2, 4), Q = (−1.04, −1.98, 3.97).
x+y+z
30. Find the local linear approximation L to the specified function f at the designated point P and compare
the error in approximating f by L at the specified point Q.
(a) f (x, y) = x0.5 y 0.3 ; P = (1, 1), Q = (1.05, 0.97),
(b) f (x, y) = x sin y ; P = (0, 0), Q = (0.003, 0.004),
2
(c) f (x, y) = p
; P = (4, 3), Q = (3.92, 3.01),
x2 + y 2
(d) f (x, y) = ln(xy) ; P = (1, 2), Q = (1.01, 2.02),
x+y
; P = (−1, 1, 1), Q = (−0.99, 0.99, 1.01),
(e) f (x, y, z) =
y+z
(f) f (x, y, z) = ln(x + yz) ; P = (2, 1, −1), Q = (2.02, 0.97, −1.01),
(g) f (x, y, z) = xyz ; P = (1, 2, 3), Q = (1.001, 2.002, 3.003),
(h) f (x, y, z) = xeyz ; P = (1, −1, −1), Q = (0.99, −1.01, −0.99).
31. A function f is given along with a local linear approximation L to f at a point P . Use the information
given to determine point P .
(a) f (x, y) = x2 + y 2 ; L(x, y) = 2y − 2x − 2,
(b) f (x, y) = x2 y ; L(x, y) = 4y − 4x + 8,
(c) f (x, y, z) = xy + z 2 ; L(x, y, z) = y + 2z − 1,
(d) f (x, y, z) = xyz ; L(x, y, z) = x − y − z − 2.
6
1.6
Maxima and Minima of Functions of Two Variables
32. Locate all relative maxima, relative minima, and saddle points, if any.
(a) f (x, y) = y 2 + xy + 3y + 2x + 3,
(b) f (x, y) = x2 + xy + y 2 − 3x,
2
(c) f (x, y) = x2 + y 2 +
,
xy
(d) f (x, y) = xey ,
(e) f (x, y) = ex sin y,
(f) f (x, y) = y sin x,
(g) f (x, y) = e−(x
2 +y 2 +2x)
(h) f (x, y) = x6 +
,
2 4
+ ,
x y
(i) f (x, y) = x2 + y − ey .
33. Find the absolute extrema of the given function on the indicated closed and bounded set R.
(a) f (x, y) = xy − x − 3y ;
(b) f (x, y) = xy − 2x ;
R is the triangular region with vertices (0, 0), (0, 4) and (4, 0),
(c) f (x, y) = xey − x2 − ey ;
(d) f (x, y) = x2 + 2y 2 − x ;
(e) f (x, y) = xy 2 ;
R is the triangular region with vertices (0, 0), (0, 4) and (5, 0),
R is the rectangular region with vertices (0, 0), (0, 1), (2, 1) and (2, 0),
R is the disk x2 + y 2 ≤ 4,
R is the region that satisfies the inequalities x ≥ 0, y ≥ 0 and x2 + y 2 ≤ 1.
34. Find three positive numbers whose sum is 48 and such that their product is as large as possible.
35. Find three positive numbers whose sum is 27 and such that the sum of their squares is as small as possible.
36. A closed rectangular box with a volume of 16 ft3 is made from two kinds of materials. The top and
bottom are made of material costing $10 per square foot and the sides from material costing $5 per square
foot. Find the dimensions of the box so that the cost of materials is minimized.
37. Find the points on the surface x2 − yz = 5 that are closest to the origin.
38. Consider the function f (x, y) = 4x2 − 3y 2 + 2xy over the unit square 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1.
(a) Find the maximum and minimum values of f on each edge of the square.
(b) Find the maximum and minimum values of f on each diagonal of the square.
(c) Find the maximum and minimum values of f on the entire square.
7
1.7
Lagrange Multipliers
39. Use Lagrange multipliers to find the maximum and minimum values of f subject to the given constraint.
Also, find the points at which these extreme values occur.
(a) f (x, y) = xy ;
4x2 + 8y 2 = 16,
(b) f (x, y) = x2 − y 2 ;
x2 + y 2 = 25,
(c) f (x, y) = 4x3 + y 2 ;
(d) f (x, y) = x − 3y − 1 ;
2x2 + y 2 = 1,
x2 + 3y 2 = 16,
(e) f (x, y, z) = 2x + y − 2z ;
(f) f (x, y, z) = 3x + 6y + 2z ;
(g) f (x, y, z) = xyz ;
x2 + y 2 + z 2 = 4,
2x2 + 4y 2 + z 2 = 70,
x2 + y 2 + z 2 = 1,
(h) f (x, y, z) = x4 + y 4 + z 4 ;
x2 + y 2 + z 2 = 1.
40. Find the point on the line 2x − 4y = 3 that is closest to the origin.
41. Find the point on the plane x + 2y + z = 1 that is closest to the origin.
42. Find the point on the plane 4x + 3y + z = 2 that is closest to (1, −1, 1).
43. Find the points on the circle x2 + y 2 = 45 that are closest to and farthest from (1, 2).
44. Find the points on the surface xy − z 2 = 1 that are closest to the origin.
45. Find a vector in 3 dimensional space whose length is 5 and whose components have the largest possible
sum.
8
บทที่ 2
Multiple Integrals
2.1
Double Integrals
1. Evaluate the iterated integrals.
1Z 2
Z
(x + 3) dy dx,
(a)
0
0
3Z 1
Z
(2x − 4y) dy dx,
(b)
−1
1
4Z 1
Z
(c)
2
0
Z
x2 y dx dy,
0
2
Z
(x2 + y 2 ) dx dy,
(d)
−2
−1
ln 3 Z ln 2
Z
(e)
0
Z
ex+y dy dx,
0
2Z 1
y sin x dy dx,
(f)
0
0
0
Z
5
Z
dx dy,
(g)
−1
2
6Z 7
Z
dy dx,
(h)
−3
4
π
Z
2
Z
x cos(xy) dy dx,
(i)
π
2
Z
1
ln 2 Z 1
(j)
0
Z
4Z
(k)
3
1
xyey
2x
dy dx,
0
2
1
dy dx.
(x + y)2
2. Evaluate the double integral over the rectangular region R.
9
ZZ
(a)
ZRZ
(b)
ZRZ
4xy 3 dA; R = {(x, y) | −1 ≤ x ≤ 1, −2 ≤ y ≤ 2},
xy
p
dA; R = {(x, y) | 0 ≤ x ≤ 1, 0 ≤ y ≤ 1},
x2 + y 2 + 1
(x sin y − y sin x) dA; R = {(x, y) | 0 ≤ x ≤
(c)
π
π
, 0 ≤ y ≤ }.
2
3
R
Z
2
Z
2
(x2 + y 2 ) dx dy.
3. Make a sketch of the solid represented by
−2
−2
4. Suppose that for some region R in the xy-plane
ZZ
f (x, y) dA = 0.
R
If R is subdivided into two regions R1 and R2 , then
ZZ
ZZ
f (x, y) dA = −
f (x, y) dA.
R1
R2
5. Show that
ZZ
f (x, y) dA =
hZ
b
g(x) dx
a
R
ih Z
d
h(y) dy
i
c
where f (x, y) = g(x)h(y) ans R = {(x, y) | a ≤ x ≤ b, c ≤ y ≤ d}.
6. Use a double integral to find the volume.
(a) The volume under the plane z = 2x + y and over the rectangle R = {(x, y) | 3 ≤ x ≤ 5, 1 ≤ y ≤ 2},
(b) The volume under the surface z = 3x3 + 3x2 y and over the rectangle R = {(x, y) | 1 ≤ x ≤ 3, 0 ≤
y ≤ 2},
(c) The volume of the solid enclosed by the surface z = x2 and the planes x = 0, x = 2, y = 3, y = 0
and z = 0.
7. The average value of a continuous function f (x, y) over a rectangle R = [a, b] × [c, d] is defined as
fave
1
=
A(R)
ZZ
f (x, y) dA
R
where A(R) = (b − a)(d − c) s the area of the rectangle R Find the average value of the following
functions over R.
(a) f (x, y) = xy 2 ; R = [0, 8] × [0, 6],
(b) f (x, y) = x2 + 7y; R = [0, 3] × [0, 6],
1
(c) f (x, y) = x(x2 + y) 2 ; R = [0, 1] × [0, 3].
10
2.2
Double Integrals over Nonrectangular Regions
8. Evaluate the iterated integral.
1Z x
Z
(a)
3
2
Z
xy 2 dy dx,
x2
0
3−y
Z
y dx dy,
(b)
1
y
√
3Z
Z
9−y 2
y dx dy,
(c)
0
0
1Z xr
Z
(d)
1
4
x2
√
Z
(e)
√
Z
2π
x3
Z
sin
π
1
0
Z
−1
dy dx,
(x2 − y) dy dx,
1Z x
y
0
y
−x2
(g)
Z
x
x2
(f)
Z
x
dy dx,
y
p
x2 − y 2 dy dx,
0
2 Z y2
e
(h)
1
(
x
)
y2
dx dy.
0
9. Let R be the region shown in the following figures. Fill in the missing limits of integration.
Z @
AZ @
A
ZZ
f (x, y) dA =
(a)
f (x, y) dy dx,
@
A
R
@
A
Z @
AZ @
A
ZZ
f (x, y) dA =
(b)
R
f (x, y) dx dy.
@
A
@
A
10. Let R be the region shown in the following figure. Fill in the missing limits of integration.
11
Z @
AZ @
A
ZZ
f (x, y) dA =
(a)
f (x, y) dy dx,
@
A @
A
Z @
AZ @
A
R
ZZ
f (x, y) dA =
(b)
f (x, y) dx dy.
@
A
R
@
A
11. Let R be the region shown in the following figure. Fill in the missing limits of integration.
Z @
AZ @
A
ZZ
f (x, y) dA =
(a)
f (x, y) dx dy,
@
A @
A
Z 2Z @
A
R
ZZ
f (x, y) dA =
(b)
1
R
Z
4Z
@
A
f (x, y) dy dx +
@
A
Z
5Z
@
A
f (x, y) dy dx +
2
@
A
f (x, y) dy dx.
4
@
A
12. Evaluate the double integral in two ways using iterated integrals: (a) viewing R as a type I region and
(b) viewing R as a type II region.
ZZ
16
(a)
x2 dA; R is the region bounded y = , y = x and x = 8,
x
R
ZZ
(b)
xy 2 dA; R is the region bounded y = 1, y = 2, x = 0 and y = x,
R
ZZ
(c)
(3x − 2y) dA; R is the region enclosed by the circle x2 + y 2 = 1,
R
ZZ
(d)
y dA; R is the region in the first quadrant enclosed between the circle x2 + y 2 = 25 and the line
R
x + y = 5,
12
ZZ
(e)
ZRZ
(f)
1
x(1 + y 2 )− 2 dA; R s the region in the first quadrant enclosed y = x2 , y = 4 and x = 0,
(x − 1) dA; R is the region in the first quadrant enclosed between y = x and y = x3 ,
ZRZ
xy dA; R is the region enclosed by y =
(g)
√
x, y = 6 − x and y = 0.
R
13. Use double integration to find the area of the plane region enclosed by the given curves.
(a) y = sin x and y = cos x for 0 ≤ x ≤
π
,
4
(b) y 2 = −x and 3y − x = 4,
(c) y 2 = 9 − x and y 2 = 9 − 9x.
14. Use double integration to find the volume of the solid.
15. Express the integral as an equivalent integral with the order of integration reversed.
Z 2 Z √x
(a)
f (x, y) dy dx,
0
0
Z 4Z 8
(b)
f (x, y) dx dy,
0
Z
2y
2 Z ey
f (x, y) dx dy,
(c)
0
Z
1
e Z ln x
f (x, y) dy dx,
(d)
1
Z
0
1Z
√
y
f (x, y) dx dy.
(e)
0
y2
16. Evaluate the integral by first reversing the order of integration.
Z 1Z 4
2
(a)
e−y dy dx,
Z0 4 Z4x2
3
ex dx dy,
(b)
√
0
Z
y
3 Z ln x
x dy dx.
(c)
1
0
13
2.3
Polar Coordinates and Applications
17. Find the rectangular coordinates of the points whose polar coordinates are given.
π
(a) (4, ),
3
π
(b) (2, − ),
6
2π
(c) (6, − ),
3
5π
(d) (4, ),
4
π
(e) (6, ),
6
2π
(f) (7, ),
3
5π
(g) (−6, − ),
6
(h) (0, −π),
17π
(i) (7,
),
6
(j) (−5, 0).
18. In each part, a point is given in rectangular coordinates. Find two pairs of polar coordinates for the point,
one pair satisfying r ≥ 0 and 0 ≤ θ < 2π and the second pair satisfying r ≥ 0 and −2π < θ ≤ 0.
(a) (−5, 0),
√
(b) (2 3, −2),
(c) (0, −2),
(d) (−8, −8),
√
(e) (−3, 3 3),
(f) (1, 1).
19. Identify the curve by transforming the given polar equation to rectangular coordinates.
(a) r = 2,
(b) r sin θ = 4,
(c) r = 3 cos θ,
(d) r = 5 sec θ,
(e) r = 2 sin θ,
(f) r = sec θ tan θ.
20. Express the given equations in polar coordinates.
14
(a) x = 3,
(b) x2 + y 2 = 7,
(c) x2 + y 2 + 6y = 0,
(d) 9xy = 4,
(e) y = −3,
(f) x2 + y 2 = 5,
(g) x2 + y 2 + 4x = 0,
(h) x2 (x2 + y 2 ) = y 2 .
21. Sketch the curve in polar coordinates.
π
,
3
3π
(b) θ = − ,
4
(c) r = 3,
(a) θ =
(d) r = 4 cos θ,
(e) r = 6 sin θ,
(f) r − 2 = 2 cos θ,
(g) r = 3(1 + sin θ),
(h) r = 4 − 4 cos θ,
(i) r = 1 + 2 sin θ,
(j) r = 4θ ; θ ≥ 0,
(k) r = 3 sin(2θ),
(l) r = 2 cos(3θ).
2.4
Double Integrals in Polar Coordinates
22. Evaluate the iterated integral.
Z π Z sin θ
2
(a)
r cos θ dr dθ,
0
Z
π
Z
0
1+cos θ
r dr dθ,
(b)
0
Z
0
π
2
Z
π
6
Z
(c)
0
Z
r2 dr dθ,
0
cos(3θ)
r dr dθ,
(d)
0
a sin θ
0
15
Z
π
1−sin θ
Z
r2 cos θ dr dθ,
(e)
0
Z
0
π
2
cos θ
Z
r3 dr dθ.
(f)
0
0
23. Express the volume of the solid described as a double integral in polar coordinates.
24. Use polar coordinates to evaluate the double integral.
ZZ
(a)
sin(x2 + y 2 ) dA where R is the region enclosed by the circle x2 + y 2 = 9,
R
ZZ p
(b)
9 − x2 − y 2 dA where R is the region in the first quadrant within the circle x2 + y 2 = 9,
R
ZZ
(c)
1
dA where R is the sector in the first quadrant bounded by y = 0, y = x and
1 + x2 + y 2
R
x2 + y 2 = 4,
ZZ
(d)
2y dA where R is the region in the first quadrant bounded above by the circle (x − 1)2 + y 2 = 1
R
and below by the line y = x.
25. Evaluate the iterated integral by converting to polar coordinates.
Z
1Z
√
1−x2
(x2 + y 2 ) dy dx,
(a)
0
Z
0
2
Z √4−y2
(b)
√
Z
0
Z
dx dy,
x2 + y 2 dy dx,
0
1Z
(d)
0
2 +y 2 )
− 4−y 2
√
2 Z 2x−x2 p
−2
(c)
e−(x
√
1−y 2
cos(x2 + y 2 ) dx dy,
0
16
Z
√
aZ
a2 −x2
(e)
dy dx
3
0
Z
(1 + x2 + y 2 ) 2
0
√
1Z
y
(f)
0
y
√
Z
p
√
2Z
0
0
(h)
−4
y
√
Z
a > 0,
x2 + y 2 dx dy,
4−y 2
(g)
Z
;
1
p
dx dy,
1 + x2 + y 2
16−x2
√
− 16−x2
3x dy dx.
26. Use a double integral in polar coordinates to find the volume of a cylinder of radius a and height h.
ZZ
27. Evaluate
x2 dA over the region R shown in the accompanying figure.
R
1
28. Show that the shaded area in the accompanying figure is a2 φ − a2 sin 2φ.
2
17
2.5
Triple Integrals
29. Evaluate the iterated integral.
1
Z
2Z 1
Z
(a)
−1
1
2
Z
0
0
π
Z
(x2 + y 2 + z 2 ) dx dy dz,
1
Z
zx sin(xy) dz dy dx,
(b)
1
3
0
0
2 Z y2
Z
z
Z
yz dx dy dz,
(c)
−1
0
π
4
Z
Z
−1
1 Z x2
x cos y dz dx dy,
(d)
0
3Z
Z
0
√
0
9−z 2
x
Z
xy dy dx dz,
(e)
0
0
0
3 Z x2
Z
ln z
Z
xey dy dz dx,
(f)
1
x
2Z
Z
√
0
4−x2
Z
3−x2 −y 2
x dz dy dx,
(g)
0
−5+x2 +y 2
0
2Z 2Z
Z
√
(h)
1
z
0
3y
x2
y
dx dy dz.
+ y2
30. Evaluate the triple integral.
ZZZ
(a)
xy sin(yz) dV where G is the rectangular box defined by the inequalities 0 ≤ x ≤ π, 0 ≤ y ≤ 1
G
π
and 0 ≤ z ≤ ,
6
ZZZ
(b)
y dV where G is the solid enclosed by the xy-plane, the plane z = y and the parabolic cylinder
G
y = 1 − x2 .
31. Use a triple integral to find the volume of the solid.
(a) The solid in the first octant bounded by the coordinate planes x ≥ 0, y ≥ 0, z ≥ 0 and the plane
3x + 6y + 4z = 12,
(b) The solid bounded by the surface z =
√
y and the planes x + y = 1, x = 0 and z = 0,
(c) The solid bounded by the surface y = x2 and the planes y + z = 4 and z = 0.
32. Let G be the solid enclosed by the surfaces in the accompanying figure. Fill in the missing limits of
integration.
18
Z @
AZ @
AZ @
A
ZZZ
f (x, y, z) dz dy dx,
f (x, y, z) dV =
(a)
@
A @
A @
A
Z @
AZ @
AZ @
A
G
ZZZ
f (x, y, z) dV =
(b)
f (x, y, z) dz dx dy.
@
A
G
@
A
@
A
33. Let G be the solid enclosed by the surfaces in the accompanying figure. Fill in the missing limits of
integration.
Z @
AZ @
AZ @
A
ZZZ
f (x, y, z) dV =
(a)
f (x, y, z) dz dy dx,
@
A @
A @
A
Z @
Z
Z
A @
A @
A
G
ZZZ
f (x, y, z) dV =
(b)
f (x, y, z) dz dx dy.
@
A
G
@
A
@
A
34. Let G be the rectangular box defined by the inequalities a ≤ x ≤ b, c ≤ y ≤ d and k ≤ z ≤ l. Show that
ZZZ
f (x)g(y)h(z) dV =
G
hZ
b
f (x) dx
a
ih Z
c
19
d
g(y) dy
ih Z
k
l
i
h(z) dz .
35. Use the result of Exercise 34 to evaluate
ZZZ
π
(a)
xy 2 sin z dV where G is the set of points satisfying −1 ≤ x ≤ 1, 0 ≤ y ≤ 1 and 0 ≤ z ≤ ,
2
G
ZZZ
e2x+y−z dV where G is the set of points satisfying 0 ≤ x ≤ 1, 0 ≤ y ≤ ln 3 and 0 ≤ z ≤ ln 2.
(b)
G
ZZZ
x + y + z dV where G is the tetrahedron shown in the following figure.
36. Evaluate the triple integral
G
37. Let G be the tetrahedron in the first octant bounded by the coordinate planes x ≥ 0, y ≥ 0, z ≥ 0 and the
plane
x y z
+ + =1
a
b
c
(a, b, c > 0)
(a) List six different iterated integrals that represent the volume of G,
1
(b) Evaluate any one of the six to show that the volume of G is abc.
6
38. Use a triple integral to derive the formula for the volume of the ellipsoid
x2 y 2 z 2
+ 2 + 2 =1
a2
b
c
20