Evaluate ZZ x2 D 1+y2dA
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Exam3Sample2.pdf
Math 252 Fall 2017 Sample Problems for Exam # 3
The relevant sections of the book are 13.1, 13.2, 13.3, 13.5, 13.6, 14.1, 12.2 (subsec- tion on arc length), 14.2
The following formulas and values will be on the �rst page of the exam:
sin �� 6
� = cos
�� 3
� = 1
2 ;
cos �� 6
� = sin
�� 3
� =
p 3
2 ;
sin �� 4
� = cos
�� 4
� =
p 2
2
Z cos2 (u)du =
u
2 + 1
2 sin(u)cos(u)Z
sin2 (u)du = u
2 � 1
2 sin(u)cos(u)
Z sink(x)dx = �
1
k sink�1(x)cos(x)+
k �1 k
Z sink�2(x)dx; k = 2;3;4; : : :Z
cosk(x)dx = 1
k cosk�1(x)sin(x)+
k �1 k
Z cosk�2(x)dx; k = 2;3;4; : : :
1. Evaluate Z Z D
x2
1+y2 dA
where D = f(x;y) : 0 � x � 1; 0 � y � 1g
(Section 13.1, problem 7).
2. Evaluate Z Z D
y2exydA
where D is the region bounded by the graphs of y = x; y = 1 and x = 0. You need to sketch the region D. (Section 13.2, problem 5).
3. Consider the iterated integralZ y=1 y=0
Z �=2 x=arcsin(y)
cos(x) p 1+cos2 (x)dxdy:
a) Sketch the region of integration D: b) Express Z Z
D
cos(x) p 1+cos2 (x)dA
1
as an iterated integral with the order of the given iterated integral reversed. You need not evaluate that iterated integral. (Based on section 13.2, problem 12).
4. Consider the double integral Z Z D
sin � x2 +y2
� dA
where D = f(x;y) : x2 +y2 � 4 and y � 0g:
a) Sketch the region D. b) Evaluate the given double integral by transforming the integral to an integral in polar coordinates. (Section 13.3, problem 2).
5. Consider the iterated integralZ y=1 y=0
Z x=p2�y2 x=y
xdxdy:
a) Sketch the region D that is relevant to the given iterated integral, b) Transform Z Z
D
xdxdy
to an iterated integral in polar coordinates. You need not evaluate that iterated integral. (Based on section 13.3, problem 11).
6. Find the volume of the region bounded by the surfaces
x2 +y2 = 4; x+y +z = 1 and z = �10:
-1 0
0
2
z
1 0
y 0
-2 2
x 0
-2
(Section 13.5, problem 2).
7. Evaluate Z Z Z D
x2dxdydz
where D is the solid tetrahedron with vertices (0;0;0) ; (1;0;0) ; (0;1;0) ; (0;0;1).
1
1
z
x
1
y
2
(Section 13.5, problem 8).
8.Evaluate Z Z Z D
x2dydxdz;
where D is bounded by the cylinder x2 +y2 = 9, the plane y+z = 5 and the plane z = 1:
x
-2
0
5
z
0
2 -2 y
0 2
(Section 13.5, problem 10).
9 Make use of cylindrical coordinates to evaluateZ Z Z D
ezdxdydz;
where D is bounded by the paraboloid z = 1+x2 +y2, the cylinder x2 +y2 = 5 and the xy-plane.
0
1
2
z
6
y 0
2
x -2 0
-2
(Section 13.6, problem 11).
10. Let D be the region between the spheres � = 2 and � = 4 and above the cone � = �=3.
0
2
4
z
y 0
42
x 0-2-4
Supply the missing data:Z Z Z D
x2zdxdydz =
Z �=? �=?
Z �=? �=?
Z �=? �=?
(?)d�d�d�
You need to express the integrand in simpli�ed form. (Based on Section 13.6, problem 26).
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11. Evaluate Z C
xeyds;
where C is the line segment traversed from (2;1) to (4;5) : (Problem 3, Section 14.2)
12. Evaluate Z C
F�d�
where F(x;y) = yi�xj
and C is the part of the unit circle traversed from (0;1) to (�1;0) in the counterclockwise direction. (Problem 7, Section 14.2)
13. Let C be the curve that is parametrized by
� (�) = (2cos(�) ;sin(�)) ; where 0 � � � �=2:
Supply the missing data: Z �C exdx+eydy =
Z �=? �=?
(?)d�
You need to express the integrand in simpli�ed form. (Based on problem 16, Section 14.2)
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