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Quiz 6 MATH 251, Section 505
Due, April, 9th, 2015
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”An Aggie does not lie, cheat or steal, or tolerate those who do”
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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
3Z 9
1. [10pts] Evaluate the integral by reversing the order of integration
0
y cos(x2 )dxdy.
y2
2. [10pts] Find the volume bounded by the paraboloid z = 5 − 2x2 − 2y 2 and the plane z = 1.
3. [10pts] Evaluate the integral
x2 + y 2 = 2 and x2 + y 2 = 3.
RR
D
xydA where D is the region of the first quadrant that lies between
2
Z
4. [10pts] Sketch the solid whose the volume is given by
0
π
2
Z
0
π
3
Z
1
ρ2 sin φdρdθdφ.
0
5. [10pts] 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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4
3
7. Let the tetrahedron be bounded by the coordinates planes and the plane x + y + z = 1 with density
ρ(x, y, z) = x .
(a) [10pts] Find the mass of this tetrahedron.
(b) [8pts] Find the x-coordinate x̄ of the center of mass of this tetrahedron.
Z
F.dr where F is the vector field F =< 3 − 8y, 4x + y > and C is the
8. Consider the line integral
C
positively oriented circle of radius 1 centered at the origin. Evaluate the integral
(a) [10pts] Directly
(b) [10pts] By Green’s Theorem
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