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Quiz 6 MATH 251, Section 506
Due, April, 2nd, 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 Z
yxdA, where D is the triangular region with vertices (0, 0), (2, 0), (2, 2) and
1. [10pts] Evaluate
D
sketch also the domain D.
RR
2. [10pts] Evaluate the integral
D xydA where D is the region of the first quadrant that lies between
x2 + y 2 = 2 and x2 + y 2 = 3 (Detail your computations). Sketch also the domain D.
Z
3Z 9
3. [10pts] Evaluate the integral by reversing the order of integration
0
y2
y cos(x2 )dxdy.
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4. [8pts] Find the volume of the solid under the paraboloid z = x2 + y 2 and above the disk x2 + y 2 ≤ 9
(use the polar coordinates).
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. [10pts] Find the total mass of a solid hemisphere of radius 2 with density ρ(x, y, z) =
1p 2
x + y2 + z2.
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7. [15pts] Evaluate the work of the force F (x, y) =< xy, −x2 − y > along the curve C given by the
vector function r(t) =< t, t2 >, 0 ≤ t ≤ 2.
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8. 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
9. [10pts] Evaluate
x2 ydx + (x − y)dy where C consists of line segments from (0, 0) to (2, 0) and from
C
(2, 0) to (3, 2) (Detail the parametrisation of C).
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