MATH 253 WORKSHEET 23
POLAR COORDINATES AND INTEGRATION
1.
Given
(x, y)
set
Polar coordinates
p
r = x2 + y 2 , θ = arctan
y
x . Given
(r, θ)
set
x = r cos θ, y = r sin θ.
D = (x, y) | 1 ≤ x2 + y 2 ≤ 2, x, y ≥ 0 .
(a) Express D in the form D = {(r, θ) | ??}
(1) Let
(b) Try expressing
(c) Calculate
Date
RR
D
RR
D
cos x2 + y 2 dA
cos x2 + y 2 dA
as an iterated integral, slicing the domain vertically.
in polar coordinates.
: 1/11/2013.
1
(2) Find the volume of the solid lying above the
the cylinder
(a) Find a
(b) Write
2
(x − 1) + y = 1.
region R in the plane
R
and
f
xy -plane,
below the paraboloid
z = x2 + y 2
and inside
2
and a function
f (x, y)
so that the volume is
RR
R
f (x, y) dA.
in polar coordinates.
(c) Evaluate the integral.
(Image credit: user Met501 and others on Wikipedia; see
to_cartesian.svg.
http://en.wikipedia.org/wiki/File:Polar_
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