MATH 2001: DOUBLE INTEGRALS IN POLAR COORDINATES
Q: Evaluate the integral
where
RR
15.4: q. 13
arctan xy dA by changing to polar coordinates,
R
R = {(x, y)|1 ≤ x2 + y 2 ≤ 4, 0 ≤ y ≤ x}
A: Noting that x2 + y 2 = r2 and that the line y = x makes an angle of
x-axis, we may rewrite this region as:
π
4
with the
n
πo
R = (x, y)|1 ≤ r ≤ 2, 0 ≤ θ ≤
4
y
To convert f (x, y) = arctan x into polar form, remember that x = rcosθ, y =
rsinθ and so tanθ = xy . In polar coordinates the integral becomes
Z Z
arctan
R
y
x
Z
π
4
2
Z
dA =
arctan(tan(θ)rdrdθ
0
Z
1
π
4
=
0
Z
=
=
=
r2
θ
2
2
1
π
4
3θ
dθ
2
0
π
3 θ2 4
2 2 0
3π 2
3 π2
=
2 32
64
References
[1] J. Stewart, Multivariable Calculus - 7th Edition, Brooks/Cole (2012).
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