10.3--Polar Functions
y
(x,y) or (r,θ)
"pole" or "origin"
r
θ
y
x
x
"polar axis" (x-axis)
Converting between polar & rectangular coordinates:
x = r cos θ
x2 + y2 = r2
y = r sin θ
tan θ =
y
x
Convert from polar coordinates to rectangular coordinates,
then sketch the graph:
1)
r = 4 cos θ
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10.3--Polar Functions
Convert from polar coordinates to rectangular coordinates,
then sketch the graph:
2)
3)
r = -6 csc θ
Find dy/dx and the slope of the tangent lines shown
on the graph of the polar equation:
π/2
r = 2 + 3 sin θ
(5, π/2)
(-1, 3π/2)
0
(-2, π)
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10.3--Polar Functions
4)
Find dy/dx at the given value of θ:
r = 3 (1 - cos θ),
θ = π/2
Solutions to dy/dθ = 0 yield horizontal tangents, provided
that dx/dθ = 0.
Solutions to dx/dθ = 0 yield vertical tangents, provided
that dy/dθ = 0.
If dy/dθ and dx/dθ are simultaneously 0, then no conclusion
can be drawn about tangent lines.
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10.3--Polar Functions
5)
Find the points of horizontal and vertical tangency
(if any) to the polar curve:
r = 1 - sin θ
special polar graphs
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10.3--Polar Functions
5
10.3--Polar Functions
Area of the entire circle = πr2
r
Area of the yellow sector =
1
2
θ r2
-Why?
β
Area of a polar region =
(from θ=α to θ=β)
π/2
1
2
r2 dθ
α
β
α
θ
0
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10.3--Polar Functions
6)
Find the area of one petal of r = 3 cos 3θ:
7)
Find the area of interior of r = 4 + 2 cos θ:
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10.3--Polar Functions
8)
9)
Find the area of the inner loop of r = 2 sin θ - 1:
Find the area between the loops of r = 2 + 4 sin θ:
Area of the outer loop =
½
(2 + 4 sin θ)2 dθ
Area of the inner loop = ½
(2 + 4 sin θ)2 dθ
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10.3--Polar Functions
10)
Find the points of intersection of the graphs of the
following equations:
r = 1- 2 cos θ
r = 1
Find the points of intersection of the following equations:
11)
r = 2 - 3 cos θ
r = cos θ
(close-up)
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10.3--Polar Functions
12)
Find the area of the common interior of r = -6 cos θ
and r = 2 - 2 cos θ
2π/3
θ=0
θ=π
θ=π
θ=0
θ = π/2
4π/3
13)
Find the area of the given region inside
r = 1 and outside r = 1 - cos θ:
10