Review Problems For Test # 3
ZZZ
Problem 1. Evaluate the integral
xyz dV, where D is the solid region bounded below
p
by xy-plane and above by the hemisphere z = 9 − x2 − y 2 .
D
ZZZ
(1 + z 2 ) dV, where D is the solid region bounded below by the
Problem 2. Evaluate
D
upper portion of the cone z 2 = 3x2 + 3y 2 and above by the sphere x2 + y 2 + z 2 = 4.
Problem 3. Sketch the solid whose volume is given by the iterated integral
Z 1 Z 4(1−x) Z 2−y2 /8
dz dy dx
0
0
0
Problem 4. Change the order of the following triple integral to dx dz dy :
Z 4 Z 2−x2 /8 Z 1−x/4
f (x, y, z) dy dz dx
0
0
0
Problem 5. Find the volume of the solid region in the first octant bounded by the paraboloid
z = x2 + y 2 and the upper portion of the cone z 2 = x2 + y 2 .
Problem 6. Find the volume of the solid region in the first octant bounded by the cylinder
x2 + y 2 = 9, the plane z = 3y, the xy−plane, and the yz−plane.
Problem
7. Evaluate the following integral, by using a suitable change of variables
RR
(2x − y 2 )dx dy, where R is the region bounded by the lines x − y = 1,
R
x − y = 3, x + y = 2, and x + y = 4.
Problem
the following integral, by using a suitable change of variables
8. Evaluate
ZZ
y−x
cos
dA, where R is the region bounded by the lines x + y = 2,
y+x
R
x + y = 4, x = 0, and y = 0.
16xy(x2 − y 2 )
dA, where R is the region in the
x2 + y 2
R
first quadrant bounded by the curves x2 + y 2 = 4, x2 + y 2 = 9, x2 − y 2 = 1, and x2 − y 2 = 4.
ZZ
Problem 9. Evaluate the following
RR
Problem 10. Evaluate the integral R (x2 + y 2 )(x2 − y 2 ) dA, where R is the region in the
first quadrant bounded by the lines y = x, y = x − 1, and circles x2 + y 2 = 1, and x2 + y 2 = 9.
Problem 11. Find the area of the region in the first quadrant bounded by x2 − y 2 = 1,
x2 − y 2 = 4, x + y = 2, and x + y = 3.
ZZ
(x + y)2 (x − y) dA, where R is the region in the
Problem 12. Evaluate the integral
R
first quadrant bounded by x2 − y 2 = 1, x2 − y 2 = 4, x + y = 2, and x + y = 3.
2
→
−
Problem 13. Is the vector field F (x, y) = h3x2 y 2 + 10x, 2x3 y + 4i conservative? If it is,
→
−
find a potential function for F .
→
−
Problem 14. Is the vector field F (x, y) = ( y 2 exy ) ~i + (1 + xy)exy ~j conservative? If it is,
→
−
find a potential function for F .
Z
Problem 15. Evaluate the line integral (2x + 3y) ds, where C is the portion of the circles
C
(x − 1)2 + y 2 = 1 from (0, 0) to (2, 0).
Z
(2x + 3y) dx, where C is the portion of the circles
√
(x − 1)2 + y 2 = 4 in the first quadrant from (3, 0) to (0, 3 ).
Problem 16. Evaluate the line integral
C
→
−
F · d~r, where C is the portion of the circles
C
→
−
(x − 1)2 + y 2 = 1 from (1, 1) to (2, 0) clockwise and F (x, y) = hy, xi.
Z
Problem 17. Evaluate the line integral
→
−
F · d~r, where C is the portion of the circles
C
→
−
(x − 1)2 + y 2 = 1 from (1, −1) to (1, 1) counter-clockwise and F (x, y) = h3x2 y + 2x, x3 i.
Z
Problem 18. Evaluate the line integral
Z
Problem 19. Evaluate the line integral
→
−
F (x, y) = h4x3 y 2 − 3y, 2x4 yi.
→
−
F · d~r, where C is the circles x2 + y 2 = 1 and
C
→
−
Problem 20. Evaluate the line integral
F · d~r, where C is the triangle formed by the
C
→
−
lines 3x + y = 3, x = 0, y = 0, and F (x, y) = h4x3 y 2 + 3y, 2x4 y + 5xi.
Z
I
Problem 21. Use Green’s theorem to evaluate
(y 3 − 2y 2 ) dx + (3y 2 x + x2 ) dy, where
C
C is the rectangle with vertices (0, 0), (3, 0), (3, 2) and (0, 2) oriented counterclockwise.