Areas and Lengths in Polar Coordinates · 2019. 12. 9. · NOTE The fact that a single point has...
Transcript of Areas and Lengths in Polar Coordinates · 2019. 12. 9. · NOTE The fact that a single point has...
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Areas and Lengths in PolarCoordinates
December 9, 2019
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Lengths in Polar Coordinates
Given a polar curve r = f(θ), we can use the relationship betweenCartesian coordinates and Polar coordinates to write parametricequations which describe the curve using the parameter θ
x(θ) = f(θ) cos(θ) y(θ) = f(θ) sin(θ)
To compute the arc length of such a curve between θ = a and θ = b,we need to compute the integral
L =
∫ b
a
√(dx
dθ
)2
+
(dy
dθ
)2
dθ
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We can simplify the formula L =
∫ b
a
√(dx
dθ
)2
+
(dy
dθ
)2
dθ because(dx
dθ
)2
+
(dy
dθ
)2
=(f ′(θ) cos(θ)− f(θ) sin(θ)
)2+(f ′(θ) sin(θ) + f(θ) cos(θ)
)2=(f ′(θ)
)2+(f(θ)
)2= r2 +
(dr
dθ
)2
= r2 + (r′)2.
L =
∫ b
a
√r2 + (r′)2 dθ
provided the parametrization moves steadily along the curve once.
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Example 1
Compute the length of the polar curve r = 6 sin θ for 0 6 θ 6 π.
I Last lecture, we saw that the graph of this equation is a circle ofradius 3 and as θ increases from 0 to π, the curve traces out thecircle exactly once.
0
Π
4
Π
2
3 Π
4
Π
5 Π
4
3 Π
2
7 Π
4
1.
2.
3.
4.
5.
6.
I We have L =
∫ π
0
√r2 + (r′)2 dθ
I r = 6 sin θ anddr
dθ= 6 cos θ. r2 + (r′)2 = 36 cos2 θ+ 36 sin2 θ = 36.
I L =
∫ π
0
√36 dθ = 6π.
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Areas in Polar Coordinates
Suppose we are given a polar curve r = f(θ) and wish to calculate thearea swept out by this polar curve between two given angles θ = aand θ = b. This is the region R in the picture on the left below:
Dividing this shape into smaller pieces (on right) and estimating theareas of the small pieces with pie-like shapes (center picture), we get aRiemann sum approximating A = the area of R of the form
A ≈n∑i=1
f(θ∗i )2
2∆θ
Letting n→∞ yields the following integral expression of the area
∫ b
a
f(θ)2
2dθ =
1
2
∫ b
a
r2dθ
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Example 2
Compute the area bounded by the curve r = sin 2θ for 0 6 θ 6 π2 .
I Using the formula A =1
2
∫ b
a
f(θ)2dθ =1
2
∫ π2
0
(sin 2θ)2dθ
I Using the half-angle formula, we get A =1
4
∫ π2
0
1− cos(4θ)dθ
=1
4
(θ − sin(4θ)
4
) ∣∣∣∣∣π/2
0
=π
8.
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Areas of Region between two curves
If instead we consider a region bounded between two polar curvesr = f(θ) and r = g(θ) then the equations becomes
1
2
∫ b
a
f(θ)2 − g(θ)2dθ
CAREFUL about parametrization!
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Example from Paul’s Online Notes
Find the area inside the circle r = 2 and the limacon r = 3 + 2 sin(θ).
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Find the area inside the circle r = 2 and the limacon r = 3 + 2 sin(θ).
The area we want is the sum of the light green and the green.
1
2
∫ 7π6
−π622dθ +
1
2
∫ 11π6
7π6
(3 + 2 sin(θ)
)2dθ
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Example 3
Find the area of the region that lies inside the circle r = 3 sin θ andoutside the cardioid r = 1 + sin θ.
I We use the formula A =1
2
∫ b
a
f(θ)2 − g(θ)2dθ
I To find the angles where the curves intersect, we find the valuesof θ where 3 sin θ = 1 + sin θ or 2 sin θ = 1. That is sin θ = 1/2
I Therefore θ =π
6,
5π
6.
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Example 3 continued
Find the area of the region that lies inside the circle r = 3 sin θ andoutside the cardioid r = 1 + sin θ.
I1
2
∫ b
a
f(θ)2 − g(θ)2dθ =1
2
∫ 5π6
π6
9 sin2 θ − (1 + sin θ)2dθ
I =1
2
∫ 5π6π6
8 sin2θ − 1 − 2 sin θdθ = 4
∫ 5π6π6
sin2θdθ −
1
2
∫ 5π6π6
1dθ −∫ 5π
6π6
sin θdθ
= 2
∫ 5π6
π6
1− cos(2θ)dθ − π
3+ cos θ
∣∣∣5π/6π/6
=4π
3− sin(2θ)
∣∣∣5π/6π/6− π
3−√
3 = π.
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Word of Warning
NOTE The fact that a single point has many representation in polarcoordinates makes it very difficult to find all the points of intersectionsof two polar curves. It is important to draw the two curves!!!Example 4 Find all possible intersection points of the curvesr = cos 2θ and r = 1
2 .
0
Π
6
Π
3
Π
22 Π
3
5 Π
6
Π
7 Π
6
4 Π
3 3 Π
2
5 Π
3
11 Π
6
1.
I Here we must solve two equations 12 = cos(2θ) and − 1
2 = cos(2θ)since the curve r = 1
2 is the same as the curve r = − 12 .
I From the first equation, we get 2θ = ±π3 , ±5π3 and from the
second we get 2θ = ± 2π3 , ±
4π3 .
I Therefore we get 8 points of intersectionθ = ±π6 , ±
π3 , ±
2π3 , ±
5π6 .
I There are infinitely many points of intersection since we may takeany of the eight points we just found and add any integralmultiple of 2π.
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Example 5
Find the area inside the circle and outside the four-leafed rose in thefirst quadrant. Equations for the curves are r = cos 2θ and r = 1
2 andwe computed the intersection points in Example 4.
From the previ-ous slide the twopoints of inter-section occur at
a =π
6and
b = −2π
3.
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r = cos(2θ)
r =1
2
Start at θ = 0 and eachpane shows a successive
interval of lengthπ
4.
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The intersection angles areπ
3and −2π
3.
1
2
∫ π4
π3
((1
4
)− cos2(2θ)
)dθ +
1
2
∫ 2π− 2π3
5π4
((1
4
)− cos2(2θ)
)dθ
OR
1
2
∫ π4
π3
((1
4
)− cos2(2θ)
)dθ +
1
2
∫ − 2π3
5π4 −2π
((1
4
)− cos2(2θ)
)dθ
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