# MATH 251-509, SPRING 2011 EXAM II - VERSION B LAST NAME (print) ```MATH 251-509, SPRING 2011
EXAM II - VERSION B
LAST NAME (print)
FIRST NAME :
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.
final answer, but also on the quality and correctness of the work leading up to it.
• SCHOLASTIC DISHONESTY WILL NOT BE TOLERATED.
THE AGGIE CODE OF HONOR
“An Aggie does not lie, cheat or steal, or tolerate those who do.”
Signature:
Good Luck!
DO NOT WRITE BELOW!
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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.
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.
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~ &middot; 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
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
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