1. hw 12 (1) Evaluate the integrals over the given intervals.
(a) Z Z
[0,1]×[0,2]
(6y2− 2x)dA.
(b) Z Z
[−1,1]×[0,π]
xy cos ydA.
(c) Z Z
[0,1]2
y
x2y2+ 1dA.
(d) Z Z Z
[0,1]3
(x2+ y2+ z2)dV.
(e) Z Z Z
[−1,1]3
(x + y + z)dV.
(f) Z 8
0
Z 2
√3
y
p1 + x4dx
! dy.
(g) Z 2
3
0
Z 1−y
2
y
(2x + y)ey−xdx
! dy
(h) Z 1
2
0
Z
√
1−y2
√3y
ln(x2+ y2)dx
! dy (i)
Z a 0
Z a x
sin(y2)dy
dx where a > 0.
(j) Z 4
0
Z 2
y 2
ex2dx
! dy.
(k) Z 2
−2
Z
√ 4−x2 0
e−(x2+y2)dy
! dx
(2) Evaluate the integrals over the given Jordan measurable regions.
(a) Z Z
R
e−(x2+xy+y2)dA where R = {(x, y) : x2+ xy + y2 ≤ 1}.
(b) Z Z
E
sin(x + y) cos(2x − y)dA where E is the region bounded by y = 2x − 1, y = 2x + 3, y = −x and y = −x + 1.
(c) Z Z
D
ex2−xy+y2dA, where D = {(x, y) : x2− xy + y2 ≤ a2}. Consider the linear transformation x = u − v/3 and y = u + v/3.
(d) Z Z
R
sin(x2+ 2xy + y2)dA where R is the region bounded by x + y = 1 and x-axis and y-axis.
(e) Z Z
D
x2
x2+ 4y2dA where D is the region bounded by x2+4y2= 1 and x2+4y2 = 4.
(f) Z Z
R
e−4x2+12xy−10y2dA where R = {(x, y) : x ≥ 2y, y ≥ 0}.
(g) Z Z
R
e−4x2−9y2dA where R = {(x, y) : 2x ≤ 3y, x ≥ 0}.
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