Final Exam:
Chapter 14-15 Book Reference: Calculus by Anton, 1oth edition
Prof. Dr. Mohammad Babul Hasan
Dept. of Mathematics
University of Dhaka, Dhaka-1000
Exercise: 14.5 (Triple Integrals)
1. Let G be the wedge in the first octant that is cut from the cylindrical solid
y2 + z2 ≤ 1 by the planes y = x and x = 0. Evaluate
 z dv .
G
2. Use a triple integral to find the volume of the solid within the cylinder x2 + y2 = 9 and
between the planes z = 1 and x + z = 5.
3. Find the volume of the solid enclosed between the paraboloids z = 5x2 + 5y2 and z = 6 − 7x2
− y2.
4. Evaluate the triple integral
 12 xy
2
z 3 dv over the rectangular box G defined by the inequalities
G
−1 ≤ x ≤ 2, 0 ≤ y ≤ 3, 0 ≤ z ≤ 2.
5. Evaluate  xyzdv where G is the solid in the first octant that is bounded by the parabolic
G
cylinder z = 2 − x2 and the planes z = 0, y = x, and y = 0.
Exercise: 14.6 (Triple Integrals in Spherical and
Cylindrical Polar Coordinates)
1. Use a triple integral to evaluate the volume of the solid enclosed between the cylinder
x 2  y 2  9 and the planes z = 1 and x + z = 5.
2
2
9 x2 9 x  y
2
3
2. Use cylindrical polar coordinate to evaluate
 
x
3  9 x 2
2
4 x2
3. Use spherical polar coordinate to evaluate  
2  4 x2
dz dy dx .
0
4 x2  y 2

z2
x 2  y 2  z 2 dz dy dx .
0
4. Use triple integration in cylindrical coordinates to find the volume of the solid G that is
bounded above by the hemisphere z  25  x 2  y 2 below by the xy-plane, and laterally by
the cylinder
x2 + y2 = 9.
5. Use spherical coordinates to find the volume of the solid G bounded above by the sphere
x2 + y2 + z2 = 16 and below by the cone z  x 2  y 2 .
Exercise: 14.7 (CHANGE OF VARIABLES IN MULTIPLE
INTEGRALS; JACOBIANS)
1. Define Jacobian of a transformation T. Let T be the transformation from the uv-plane to the
xy- plane and is defined be the equations
(i)
(ii)
(iii)
x
1
4
u  v  and
y
1
4
u  v  . Find the followings:
T(3, 1)
Sketch the u-curves corresponding to v = -2, -1, 0, 1, 2
Sketch the v-curves corresponding to u = -2, -1, 0, 1, 2
2. Use Jacobian to evaluate   x  y  e x
2
 y2
dA
over the rectangle
R
R  x  y  0, x  y  1, x  y  1, x  y  4
3. Use Jacobian to evaluate 
R
x y
x y
dA
over the rectangle
R  x  y  0, x  y  1, x  y  1, x  y  3
4. Use Jacobian to evaluate  e xy dA over the region R enclosed by rectangle
R
y
x
2
, y  x, y 
1
x
,y
2
x
.
Chapter 15
(15.1, 15.2, 15.4, 15.5, 15.7, 15.8)
Vector Calculus
Divergence of F: If F(x, y, z) = f (x, y, z)i + g(x, y, z)j + h(x, y, z)k, then we define the divergence
of F, written div F, to be the function given by
div F = ∂f /∂x + ∂g /∂y + ∂h /∂z
Curl of F: If F(x, y, z) = f (x, y, z)i + g(x, y, z)j + h(x, y, z)k, then we define the curl of F, written
curl F, to be the vector field given by
curl F = (∂h/ ∂y − ∂g /∂z) i + (∂f /∂z − ∂h/ ∂x) j + (∂g /∂x − ∂f/ ∂y) k
Curl
F 
i

j

k

x
f
y
g
z
h
Example 1: Find the divergence and the curl of the vector field F(x, y, z) = x2 yi + 2y3 zj + 3zk
Example 2: Show that the divergence of the inverse-square field F(x, y, z) = c /(x2 + y2 + z2)3/2 *
(xi + y j + zk) is zero
Line integral (15.2)
Line integral: If C is smoothly parametrized by r(t) = x(t)i + y(t)j (a ≤ t ≤ b) then

f  x , y ds 
C
b
 f  x t , y t  r t 
dt
a
Similarly, if C is a curve in 3-space that is smoothly parametrized by r(t) = x(t)i + y(t)j + z(t)k (a ≤ t ≤
b
b) then
 f  x , y , z ds   f  x t , y t , z t  r t 
C
dt
a
Example 1: Using the given parametrization, evaluate the line integral
 1  xy ds
2
C
(a) C : r(t) = ti + 2tj (0 ≤ t ≤ 1)
(b) C : r(t) = (1 − t)i + (2 − 2t)j (0 ≤ t ≤ 1)
Alternative formula expression for a curve C in the xy-plane that is given by parametric equations x =
x(t), y = y(t) (a ≤ t ≤ b)

f  x , y ds 
c
b

a
2
2
 dx 
 dy 
f  x t , y t  
 
 dt
 dt 
 dt 
Example: Evaluate   x  2 y  dx  x 2  y 2 dy along the arc C given the line segment (0, 0) to (1, 0)
C
and (1, 0) to (1, 1).
Example: Evaluate
 xy dx   x  y dy along the arc C given the line segment
C
(i)
(ii)
(iii)
(iv)
(0, 0) to (1, 0) and (1, 0) to (1, 3).
(0, 0) to (0, 3) and (0, 3) to (1, 3)
(0, 0) to (1, 3)
C is the graph of y  3 x 2 from (0, 0) to (1, 3).
Example: Evaluate
 xy dx  x
2
dy along the arc C given the line segment
C
(i)
(ii)
(iii)
(2, 1) to (4, 1) and (4, 1) to (4, 5).
(2, 1) to (4, 5)
C is the graph of x  3t  1, y  3t 2  2t , 1  t  5 / 3 .
Example: Evaluate
 x
C
2

 y 2 dx  2 xy dy along the arc C x  t 2  1 , y  t 2  t  2 ,
0  t  1.
Example: Evaluate the line integral
 xy  z ds from(1, 0, 0)to (−1, 0, π) along the helix C that is
3
C
represented by the parametric equations x = cost, y = sin t, z = t (0 ≤ t ≤ π)
Example: Evaluate  3 xy dy where C is the line segment joining (0, 0) and (1, 2) with the given
C
orientation. (a) Oriented from (0, 0) to (1, 2)
Example: Evaluate
 2 xy dx  x
2
(b) Oriented from (1, 2) to (0, 0)

 y 2 dy along the circular arc C given by x = cost, y = sin t (0 ≤ t ≤
C
π/2)
Example: Evaluate  2 xy dy  3 x 2  y 2 dx along the circular arc C given by x = cost, y = sin t (0 ≤ t
C
≤ π/2)
Definition: If F is a continuous vector field and C is a smooth oriented curve, then the line integral of
F along C is  F .dr
C
Suppose that C is an oriented curve in the plane given in vector form by r = r(t) = x(t) i + y(t) j (a ≤ t
≤ b). If we write F(r(t)) = f (x(t), y(t))i + g(x(t), y(t))j then
b
Example: Evaluate
 F .dr 
 F r t .r t 
C
a
dt
 F .dr where F(x, y) = cos xi + sin xj and where C is the given oriented curve. (a)
C
C : r(t) = −π 2 i + tj (1 ≤ t ≤ 2)
Example: evaluate
(b) C : r(t) = ti + t 2 j (−1 ≤ t ≤ 2)
 F .dr where F(x, y) = −yi + xj and where C is the given oriented curve. (a) C :
C
x2 + y2 = 3 (0 ≤ x, y; anticlockwise) (b) C : r(t) = ti + 2tj (0 ≤ t ≤ 1;
Green’s Theorem (15.4)
Green’s Theorem: Let R be a simply connected plane region whose boundary is a simple, closed,
piecewise smooth curve C oriented counterclockwise. If f (x, y) and g(x, y) are continuous and have
continuous first partial derivatives on some open set containing R, then

f  x , y  dx  g  x , y dy 
C
 g f 
  x  y  dA
R
Example 1: Use Green’s Theorem to evaluate
x
2
y dx  x dy along the triangular path (0,0) to
(1,
C
0), (1, 0) to (1, 2) and (1, 2) to (0, 0).
Example 2: Find the work done by the force field F(x, y) = (ex − y3 )i + (cos y + x3 )j on a particle
that travels once around the unit circle x2 + y2 = 1 in the counterclockwise direction.
Example 3: Verify Green’s theorem for
 2 cy  x  dx  x  y  dy where C is the closed curve of
2
2
C
the region bounded by y  x and x  y 2 .
2
Example 4: Verify Green’s theorem for
x
2
y dx  x dy where C is the line segment from (0, 0) to
C
(1, 0) to (1, 2) to (0, 0).
Example 5: Verify Green’s theorem for  3 xy dx  2 xy dy where C is the rectangle enclosed by
C
= - 2, x = 4, y = 1, y = 2.
x
Surface integral (15.5)
Surface integral: Let σ be a smooth parametric surface whose vector equation is r = x(u, v)i + y(u,
v)j + z(u, v)k where (u, v) varies over a region R in the uv-plane. If f (x, y, z) is continuous on σ,
then
r
r
 f  x, y , z  ds   f  x u , v , y u , v , z u , v  u  v

dA
R
Example: Evaluate the surface integral
 x
2
ds over the sphere x2 + y2 + z2 = 1.

(a) Let σ be a surface with equation z = g(x, y) and let R be its projection on the xyplane. If g has
continuous first partial derivatives on R and f (x, y, z)is continuous on σ, then
 f  x , y , z  ds   f  x , y , g  x , y 

R
2
2
 z 
 z 
  1

  
 x 
 y 
dA
(b) (b) Let σ be a surface with equation y = g(x, z) and let R be its projection on the xzplane. If g
has continuous first partial derivatives on R and f (x, y, z)is continuous on σ, then

f  x , y , z  ds 

2

R
2
 y 
 y 
f  x , g  x , y , z  
 
 1
 x 
 z 
dA
(c) Let σ be a surface with equation x = g(y, z) and let R be its projection on the yzplane. If g has
continuous first partial derivatives on R and f (x, y, z)is continuous on σ, then
2
 f  x , y , z  ds   f  g  x , y , y , z 

R
Example: Evaluate the surface integral
2
 x 
 x 

     1
 z 
 y 
dA
 xz ds where σ is the part of the plane x + y + z = 1 that lies

in the first octant
Example: Evaluate the surface integral
 y
2
z 2 ds where σ is the part of the cone z = x2 + y2 that

lies between the planes z = 1 and z = 2
Example: Suppose that a curved lamina σ with constant density δ(x, y, z) = δ0 is the portion of the
paraboloid z = x2 + y2 below the plane z = 1. Find the mass of the lamina.
Example: Evaluate
   x , y , z  ds where  is the surface of the paraboloid

z  2  ( x  y )  f  x , y  above the xy-plane and   x , y , z   x 2  y 2
2
2
Example: Evaluate
 x
2
z ds where  is the portion of the cone z 2  x 2  y 2 lying between z =1
 x
2
2
2
2
2
ds where  is the upper half of the sphere a  x  y  z .

and z = 4.
Example: Evaluate

The Divergence Theorem (15.7)
The Divergence Theorem: Let G be a solid whose surface σ is oriented outward. If F(x, y, z) = f (x,
y, z)i + g(x, y, z)j + h(x, y, z)k where f, g, and h have continuous first partial derivatives on some
open set containing G, and if n is the outward unit normal on σ, then
 F . n ds   divF

dV
G
Example: Use the Divergence Theorem to find the outward flux of the vector field F(x, y, z) = zk
across the sphere x2 + y2 + z2 = a2.
Example: Use the Divergence Theorem to find the outward flux of the vector field F(x, y, z) = 2xi +
3y j + z2 k across the unit cube.
Example: Use the Divergence Theorem to find the outward flux of the vector field F(x, y, z) = x3 i +
y3 j + z2 k across the surface of the region that is enclosed by the circular cylinder x2 + y2 = 9 and
the planes z = 0 and z = 2.
Example: Use the Divergence Theorem to find the outward flux of the vector field F(x, y, z) = x3 i
+ y3 j + z3 k across the surface of the region that is enclosed by the hemisphere z  a 2  x 2  y 2
and the plane z = 0.
Example: Let V be the region bounded by x 2  y 2  4 ,
z  0,
z  3 . S is the surface of V. If
 A . n ds .
A ( x , y , z )  x 3 i  y 3 j  z 3 k . Use divergence theorem to find
s
Example: Let V be the region bounded by z  4  x 2 ,
y  z  5 and xy and xz plane. S is the
surface of V. If A ( x , y , z )  x 3  sin z i  x 2 y  cos z  j  e x
 A . n ds
s
2
 y2
k . Use divergence theorem to find
Stokes’ Theorem (15.8)
Stokes’ Theorem: Let σ be a piecewise smooth oriented surface that is bounded by a simple, closed,
piecewise smooth curve C with positive orientation. If the components of the vector field F(x, y, z) =
f (x, y, z)i + g(x, y, z)j + h(x, y, z)k are continuous and have continuous first partial derivatives on
some open set containing σ, and if T is the unit tangent vector to C, then
 F. T
ds 
C
It can also be written as
 F . dr   CurlF
C
 CurlF
. n ds

. n ds

Example: Find the work performed by the force field F(x, y, z) = x2 i + 4xy3 j + y2 xk on a particle
that traverses the rectangle C in the plane z = y
Example: Verify Stokes’Theorem for the vector field A  y 2 i  z 2 j  x 2 k . And S is the first octant
portion of the plane x  y  z  1 .
Example: Verify Stokes’Theorem for the vector field A  2 y i  3 j  3k . And S is the part of the
papaboloid z  4  x 2  y 2 which lies inside the cylinder x 2  y 2  1 .
Example: Verify Stokes’Theorem for the vector field A  3 z i  4 xj  2 yk . And S is the part of the
papaboloid z  9  x 2  y 2 .
Example: Verify Stokes’Theorem for the vector field F(x, y, z) = 2zi + 3x j + 5yk, taking σ to be the
portion of the paraboloid z = 4 − x2 − y2 for which z ≥ 0 with upward orientation, and C to be the
positively oriented circle x2 + y2 = 4 that forms the boundary of σ in the xy-plane.