Area in Polar Coordinates
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Transcript of Area in Polar Coordinates
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Area in Polar Coordinates
Objective: To find areas of regions that are bounded by polar
curves.
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Area of Polar Coordinates• We will begin our investigation of area in polar
coordinates with a simple case.
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Area of Polar Coordinates• In rectangular coordinates we obtained areas under
curves by dividing the region into an increasing number of vertical strips, approximating the strips by rectangles, and taking a limit. In polar coordinates rectangles are clumsy to work with, and it is better to divide the region into wedges by using rays.
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Area of Polar Coordinates• As shown in the figure, the rays divide the region R
into n wedges with areas A1, A2,…An and central angles . The area of the entire region can be written as
n
kkn AAAAA
121 ...
n ,..., 21
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Area of Polar Coordinates• If is small, then we can approximate the area
of the kth wedge by the area of a sector with central angle and radius where is any ray that lies in the kth wedge. Thus, the area of the sector is
kAk
k )( *kf *
k
k
n
kk
n
kk fAA
2
1
*21
1
])([
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Area of Polar Coordinates• If we now increase n in such a way that max ,
then the sectors will become better and better approximations of the wedges and it is reasonable to expect that the approximation will approach the exact value.
0 k
dffA k
n
kk
k
2212
1
*21
0max)]([])([lim
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Area of Polar Coordinates• This all leads to the following.
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Area of Polar Coordinates• The hardest part of this is determining the limits of
integration. This is done as follows:
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Example 1• Find the area of the region in the first quadrant that
is within the cardioid r = 1 – cos.
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Example 1• Find the area of the region in the first quadrant that
is within the cardioid r = 1 – cos.• The region and a typical radial line are shown. For
the radial line to sweep out the region, must vary from 0 to /2. So we have
1)cos1( 83
2/
0
221
2/
0
221
ddrA
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Example 2• Find the entire area within the cardioid r = 1 – cos.
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Example 2• Find the entire area within the cardioid r = 1 – cos.• For the radial line to sweep out the entire cardioid,
must vary from 0 to 2. So we have
23)cos1(
2
0
221
2
0
221
ddrA
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Example 2• Find the entire area within the cardioid r = 1 – cos.• For the radial line to sweep out the entire cardioid,
must vary from 0 to 2. So we have
• We can also look at it this way.
23)cos1(2
0
2
0
221
ddrA
23)cos1(
2
0
221
2
0
221
ddrA
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Example 3• Find the area of the region enclosed by the rose curve
r = cos2.
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Example 3• Find the area of the region enclosed by the rose curve
r = cos2.• Using symmetry, the area in the first quadrant that is
swept out for 0 < < /4 is 1/8 of the total area.
22cos48
4/
0
24/
0
221
ddrA
22cos
2
0
221
2
0
221
ddrA
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Example 4• Find the area of the region that is inside of the
cardioid r = 4 + 4 cos and outside of the circle r = 6.
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Example 4• Find the area of the region that is inside of the
cardioid r = 4 + 4 cos and outside of the circle r = 6.• First, we need to find the bounds.
21cos
cos446
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Example 4• Find the area of the region that is inside of the
cardioid r = 4 + 4 cos and outside of the circle r = 6.• The area of the region can be obtained by
subtracting the areas in the figures below.
4318)6()cos44(3/
3/
221
3/
3/
221
ddA
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Example 5• Find the area of the region that is common to the
two regions bounded by the cardioid and the circle r = -6cos.
cos22 r
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Example 5• Find the area of the region that is common to the
two regions bounded by the cardioid and the circle r = -6cos.
cos22 r
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Homework
• Pages 726-727• 1-9 odd• 13, 19, 21, 25