1. hw 4 (1) Read Section 15.1 and 15.2 of the Book by Stewart.
(2) Express the following limit as an integral.
(a) lim
n→∞
1 n
n
X
k=1
r 1 + k
n. (b) lim
n→∞
√1 n
1
√n + 1+ 1
√n +√
2+ · · · + 1
√n +√ n
. (3) Express the following limit as a double integral
m,n→∞lim 1 nm
n
X
i=1 m
X
j=1
sin 2πi n +3πj
m
.
(4) Suppose that we know
cos 3θ = 4 cos3θ − 3 cos θ sin 3θ = 3 sin θ − 4 sin3θ (sin ax)0= a cos(ax) (cos ax)0= −a sin(ax).
Evaluate the following double integrals:
(a) Z Z
[0,3]×[0,π/2]
x2sin3ydA.
(b) Z Z
[0,1]×[0,π/2]
(x2sin y + x cos3y)dA.
(c) Z Z
[0,π/6]×[0,π/2]
(sin x + sin y)dA.
(5) Suppose we know that
(tan x)0= sec2x, (sec x)0= sec x tan x.
Evaluate the following double integrals (a)
Z Z
[0,1]×[0,π/3]
x sec y tan ydA.
(b) Z Z
[0,π/4]×[0,π/2]
sec2x sin ydA.
(6) Let D be the plane region bounded by y = 2x2and y = 1 + x2. Evaluate the double integral Z Z
D
1dA.
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