MATH 251-509, SPRING 2011
EXAM II - VERSION B
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UIN:
SEAT#:
DIRECTIONS:
• The use of a calculator, laptop or computer is prohibited.
• In all problems present your solutions in the space provided.
• Be sure to read the instructions to each problem carefully.
• Use a pencil and be neat. If I can’t read your answers, then I can’t give you credit.
• Show all your work and clearly indicate your final answer. You will be graded not merely on the
final answer, but also on the quality and correctness of the work leading up to it.
• SCHOLASTIC DISHONESTY WILL NOT BE TOLERATED.
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“An Aggie does not lie, cheat or steal, or tolerate those who do.”
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100
GRADE:
1
1. Given
−2 Z x+6
Z
ZZ
−6
D
Z
0
Z
x2
f (x, y) dydx.
f (x, y) dydx +
f (x, y) dA =
−2
0
0
(a) [4pts] sketch the region of integration D.
y
x
0
(b) [8pts] change the order of integration.
WRITE YOUR ANSWER HERE:
2. [10pts] Convert the integral
Z
1
−1
Z √1−y2 Z
0
√
0
−
ln(4 + x2 + y 2 + z 2 ) dz dx dy
1−x2 −y 2
to an integral in spherical coordinates, but don’t evaluate it.
WRITE YOUR ANSWER HERE:
2
3. [12pts] Find the mass of the lamina that occupies the region D = (x, y) : x2 + y 2 ≤ 2x, y ≥ 0
and has the density ρ(x, y) = y.
4. [10pts] Find themass of a thin wire in the shape of C with the density ρ(x, y, z) = 7y 2 z if C is
2 3
given by r(t) =
t , t, t2 , 0 ≤ t ≤ 1.
3
Hint: (a + b)2 = a2 + b2 + 2ab
3
5. [12pts] Find the volume of the paraboloid z = x2 + y 2 below the paraboloid z = 8 − 3(x2 + y 2 ).
4
6. [10pts] Find the work done by the force field F(x, y) = 5 + y, − 13 x on a particle that moves along
the curve y = x3 from (−1, −1) to the point (1, 1).
I
7. [10pts] Use Green’s Theorem to compute the integral 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).
5
8. Let F~ (x, y) = hx3 y 4 , x4 y 3 i.
a) [2pts] Show that F~ is conservative vector field.
Z
b) [10pts] Compute
F~ · d~r where C is any path from the point M(0,0) to the point N(1,2).
C
6
9. [12pts] Let f (x, y) = 3y − 2xy + 1. Find the absolute maximum and minimum values of f on the
region D bounded by the curves y = x2 and y = 3x.
7
Answers
Z
4Z
√
− y
1b
dxdy
0
y−6
3 2/3
4 17/5
5 8π
6 10
9 min f = f (3, 9) = −26, max f = f (3/4, 9/4) = 35/8
8