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Home Quiz 2 (Quiz 6), MATH 251, Section 505
Due, November, 12th, 2015
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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
2 Z ln x
1. [8pts] Evaluate the integral by reversing the order of integration
1
x2 + y dy dx.
0
2
2
2. [15pts] Find the
p volume of the solid E in the first octant bounded by the paraboloid z = x + y and
2
2
the cone z = x + y 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
2
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 total mass of the lamina having the density ρ(x, y) = xy and occupying the region D
which is in the first quadrant bounded by y = x2 and y = 1.
6. [15pts] Find the area enclosed by the ellipse
x2
+ y 2 = 1 (Hint : Use Green’s Theorem).
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7. Let the tetrahedron be bounded by the coordinates planes and the plane 2x + y + 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
F.dr where F is the vector field F =< 4 − 6y, 2x + y > and C is the
8. Consider the line integral
C
positively oriented circle of radius 2 centered at the origin. Evaluate the integral (Show your work)
(a) [10pts] Directly
(b) [10pts] By Green’s Theorem
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