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Home Quiz 2, MATH 251, Section 505
Due April, 7th 2016
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first name : . . . . . . . . . . . . . . . . . .
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Write up your result, detail your calculations if necessary and BOX your final answer.
Your final work have to be neat, so use pencil first if you want.
Z
8Z 2
1. [8pts] Evaluate the integral by reversing the order of integration
0
1
y3
p
x4 + 1 dx dy.
2. [15pts] Find the volume of the solid E in the first octant bounded by the paraboloid z = 12 (x2 + y 2 )
p
and the cone z = 2x2 + 2y 2 and coordinate planes.
3. [10pts] Evaluate the integral
x2 + y 2 = 1 and x2 + y 2 = 2.
RR
D
xydA where D is the region of the first quadrant that lies between
The domain D is a part of an annulus, described in cartesian coordinates by
{(x, y) | 1 ≤ x2 + y 2 ≤ 2, x ≥ 0, y ≥ 0}
and in polar coordinates by
{(r, θ) | 1 ≤ r ≤
√
2, 0 ≤ θ ≤
π
}.
2
2
So, we obtain
Z
Z Z
xydA =
D
π
2
0
r4
=
4
=
√
Z
1
√
2
r3 cos θ sin θdr dθ,
2
1
cos2 θ
−
2
π2
,
0
3
8
Z
π
Z
4. [8pts] Sketch the solid whose the volume is given by
0
0
π
4
Z
1
ρ2 sin φdρdθdφ.
0
5. [8pts] Find the volume of the solid E bounded by the paraboloid z = 2 − x2 − y 2 and the paraboloid
z = 1 + x2 + y 2 .
6. [15pts] Find the area enclosed by the ellipse
x2 y 2
+
= 1 (Hint : Use Green’s Theorem).
2
3
3
7. Let the tetrahedron be bounded by the coordinates planes and the plane 3x + 3y + z = 1 with density
ρ(x, y, z) = y .
(a) [8pts] Find the mass of this tetrahedron.
(b) [8pts] Find the y-coordinate ȳ of the center of mass of this tetrahedron.
Z
8. Consider the line integral
F.dr where F is the vector field F =< 4x2 − 6y, 4x + y 2 > and C is the
C
positively oriented circle of radius 3 centered at the origin. Evaluate the integral (Show your work)
(a) [10pts] Directly
(b) [10pts] By Green’s Theorem
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