MathB6C / Fall 2009
Integral Practice (Chapter 15)
Remark: Corrections have been made to problems #4, 10, and 17.
Evaluate the following:
2
2
1)
∫ ∫ y + 2xe
2)
∫ ∫ ye
3)
4)
5)
1
0
1
1
0
0
1
1
0
y
1
ex
0
x
xy
dxdy
dxdy
∫ ∫ cos( x
∫∫
y
2
) dxdy
3xy 2 dydx
1
dA
1+ x2
where D is bounded by the triangular region with vertices (0,0), (1,1), (0,1).
∫∫
6)
∫∫
7)
∫∫
D
D
D
xy dA , D = {(x, y) │0 ≤ y ≤ 1, y2 ≤ x ≤ y + 2}
y dA where D is the region in the first quadrant that lies above the hyperbola xy =1
and the line y = x and below the line y = 2.
8)
∫ ∫ (x
2
D
+ y2
and y =
9)
∫∫
D
)
3/ 2
dA where D is the region in the first quadrant bounded by the lines y = 0
3 x and the circle x2 + y2 = 9.
x dA where D is the region in the first quadrant that lies between the circles
x2 + y2 = 1 and x2 + y2 = 2.
10)
3
∫ ∫
−3
0
9−x 2
sin( x 2 + y 2 ) dydx
11)
∫∫
xy dA
D
where D is the region bounded by the line y = x – 1 and the parabola y2 = 2x + 6.
π
1
0
0
12)
∫ ∫ ∫
13)
∫∫ ∫
W
1− y 2
y sin x dzdydx
0
xy dV
where W is the solid tetrahedron with vertices (0,0,0), (1/3, 0,0), (0,1,0), and (0,0,1).
14)
∫∫∫
15)
∫∫ ∫
W
y 2 z 2 dV where W is bounded by the paraboloid x = 1 – y2 – z2 and the plane x = 0.
W
yz dV where W lies above the plane z = 0, below the plane z = y, and inside the
cylinder x2 + y2 = 4.
16)
∫∫∫
W
z 3 x 2 + y 2 + z 2 dV where W is the solid hemisphere that lies above the xy-plane
and has center the origin and radius 1.
17)
4− y 2
2
∫ ∫
−2
0
∫
4− x 2 − y2
− 4− x 2 − y 2
y 2 x 2 + y 2 + z 2 dzdxdy
Find the following Volumes
18) The tetrahedron bounded by the planes x + 2y + z = 2, x = 2y, x = 0, and z = 0.
19) Under the plane x + 2y – z = 0 and above the region bounded by y = x and y = x4.
20) Under the surface z = 2x + y2 and above the region bounded by x = y2 and x = y3.
21) Enclose by the cylinders z = y2, y = x2, and the planes z = 0 and y = 4.
22) Bounded by the cylinder x2 + y2 = 1 and the planes y = z, x = 0, z = 0 in first octant.
23) Bounded above by the cone z = x 2 + y 2 and below x2 + y2 + z2 = 16.
Find the Mass
24) Find the mass of the solid tetrahedron with vertices (0,0,0), (1,0,0), (0,2,0), (0,0,3) and
density function ρ(x,y,z) = x2 + y2 + z2.
25) Find the mass of a thin plate of density ρ(x,y) = y bounded by the parabola x = 1 – y2 and
the coordinate axis in first quadrant.
Jacobian
26)
27)
x − 2y
dA where D is the parallelogram enclosed by the lines
3x − y
x – 2y = 0, x – 2y = 4, 3x – y = 1, and 3x – y = 8.
∫∫
∫∫
D
D
sin(9 x 2 + 4 y 2 ) dA where D is the region in first quadrant bounded by the ellipse
9x2 + 4y2 = 1.