MATH 251
REVIEW
EXAM 3
CONCEPTS TO KNOW
(Review all the concepts below and only then work practice problems )
• Triple integral (13.8);
• Volume of a solid by triple integral (13.8);
• Mass of a solid (13.8);
• Cylindrical and Spherical coordinates (13.9);
• Triple integral in cylindrical and spherical coordinates (13.10);
• Conservative vector field: definition (14.3);
• Fundamental Theorem of Calculus (14.3);
• Green’s Theorem and applications (14.4);
• Line integral of a conservative vector field: independence of path; integration over
closed path (14.3);
• Work done by conservative force field; potential function (14.3);
• Parametric surface, parameters domain, normal vector to parametric surface, tangent plane to parametric surface. (14.6);
• Surface area of a parametric surface and surface area of a graph z = f (x, y)(14.6);
• Surface integral of a scalar function, mass of a thin sheet (14.7);
• Oriented surface (14.7);
• Surface integral of a vector field, flux (14.7).
Formulas sheet you may use during the test (will be typed on the title page of the
test):
x = ρ sin φ cos θ, y = ρ sin φ sin θ, z = ρ cos φ
I
P dx + Q dy =
ZZ
∂D
D
∂Q ∂P
−
∂x
∂y
dS = |N(u, v)|dA
dS = n̂ dS
1
!
dA.
Practice Problems
1. Write the equation x2 + y 2 + z 2 = 2x in cylindrical coordinates and in spherical
coordinates.
2. Evaluate
Z Z
F · dS for the vector field F(x, y, z) = hxz, yz, z 2 i over the circular
S
cylinder x2 + y 2 = 9 for between the planes z = 0 and z = 2 oriented outward.
3. Evaluate
Z Z
F · dS for the vector field F(x, y, z) = hz, z, x + yi over the surface
S
S : r(u, v) = hu + v, u − v, uvi for 0 ≤ u ≤ 2 and 0 ≤ v ≤ 3.
4. Let F~ (x, y) = hx + y 2 , 2xy + y 2 i.
a) Show that F~ is conservative vector field.
b) Compute
Z
F~ · d~r where C is any path from (-1,0) to (2,2).
C
√
Z 3Z
9−y 2
√
Z
9−x2 −y 2
5. Convert the integral
(x2 + y 2 + z 2 ) dz dx dy to an integral in
0
0
0
spherical coordinates, but don’t evaluate it.
6. Find the volume of the solid
E in the first octant bounded by the paraboloid
√ region
2
2
2
2
z = x + y , the cone z = x + y and the coordinate planes.
7. Use Green’s Theorem to compute the integral
Z
(12 − x2 y − y 3 + tan x) dx + (xy 2 + x3 − ey ) dy
C
where C is positively oriented boundary of the region enclosed by the circle x2 +y 2 =
4. Sketch the curve C indicating the positive direction.
8. Let F(x, y) = h2x + y 2 + 3x2 y, 2xy + x3 + 3y 2 i.
(a) Show that F is conservative vector field.
(b) Evaluate
Z
F · dr where C is the arc of the curve y = x sin x from (0, 0) to
C
(π, 0).
(Hint: use the previous part.)
9. Find the mass of the tetrahedron with vertices (0, 0, 0), (1, 0, 0), (0, 2, 0) and (0, 0, 4),
if the density is ρ = x.
√
10. Find the volume of the cone z = 2 x2 + y 2 below the paraboloid z = 8 − x2 − y 2 .
2
11. Convert the integral
√
Z 1 Z
−1
1−y 2
Z 0
√
−
0
1−x2 −y 2
ln(4 + x2 + y 2 + z 2 ) dz dx dy
to an integral in spherical coordinates, but don’t evaluate it.
12. Use Green’s Theorem to compute the integral I =
I
(cos x4 + xy)dx + (y 2 ey + x2 )dy,
C
where C is the triangular curve consisting of the line segments from (0, 0) to (1, 0),
from (1, 0) to (1, 3), and from (1, 3) to (0, 0).
13. Let F~ (x, y) = hx3 y 4 , x4 y 3 i.
a) Show that F~ is conservative vector field.
b) Compute
Z
F~ · d~r where C is any path from the point M(0,0) to the point
C
N(1,2).
1
2
2
2
2
14. Compute
S x dS on the parametric surface r(u, v) = hu + v , u − v , 2uvi for
1 ≤ u ≤ 3 and 1 ≤ v ≤ 4.
RR
15. Find the mass of the 1/8 of the solid sphere x2 + y 2 + z 2 ≤ 16 in the first octant if
the mass density is ρ(x, y, z) = z.
16. Sketh the solid consisting of all points with spherical coordinates (ρ, θ, φ) such that
0 ≤ θ ≤ π/2, 0 ≤ φ ≤ π/6 and 0 ≤ ρ ≤ 2 cos φ.
17. (a) Find an equation of the tangent plane at the point (4, −2, 1) to the parametric
surface S given by r(u, v) = hv 2 , −uv, u2 i, 0 ≤ u ≤ 3, −3 ≤ v ≤ 3.
(b) Set up, but do not evaluate an (iterated) integral for the surface area of S.
18. Convert the integral
√
Z 1Z
0
0
1−y 2
√
x2 +y 2
Z
x2 +y 2
xyz dz dx dy
to an integral in cylindrical coordinates, but don’t evaluate it.
19. Use Green’s Theorem to compute the integral along the given positively oriented
curve C:
Z
(y 2 − arctan x) dx + (3x + sin y) dy,
C
where C is the boundary of the region enclosed by the parabola y = x2 and the line
y = 4.
3
Answers
1. z 2 = r(2 cos θ − r); ρ = 2 sin φ cos θ
2. 36π
3. 12
4. (b)73/6
5.
Z π/2 Z π/2 Z 3
ρ4 sin φ dρ dφ dθ
0
0
0
6. π/24
7. 32π
8. (b)π 2
9. 1/3
10.
11.
40π
3
Z π/2 Z π Z 1
−π/2
π/2
ln(4 + ρ2 )ρ2 sin φ dρ dφ dθ
0
12. 1
13. (b) 4
√
14. 24 2
15. 16π
16. solid in the first octant bounded by sphere (with radius 1 and center (0, 0, 1)) and
above the cone φ = π/6.
17. (a) x
+ 4y + 4z = 0;
Z 3Z 3 √
(b)
2 u4 + 4u2 v 2 + v 4 dv du
−3
0
18.
Z π/2 Z 1 Z r
0
0
r2
r3 z cos θ sin θ dz dr dθ
19. −96/5
4