area in polar coordinates
DESCRIPTION
Area in Polar Coordinates. Objective: To find areas of regions that are bounded by polar curves. Area of Polar Coordinates. We will begin our investigation of area in polar coordinates with a simple case. Area of Polar Coordinates. - PowerPoint PPT PresentationTRANSCRIPT
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Area in Polar Coordinates
Objective: To find areas of regions that are bounded by polar
curves.
![Page 2: Area in Polar Coordinates](https://reader031.vdocuments.site/reader031/viewer/2022033021/56813679550346895d9e083a/html5/thumbnails/2.jpg)
Area of Polar Coordinates
• We will begin our investigation of area in polar coordinates with a simple case.
![Page 3: Area in Polar Coordinates](https://reader031.vdocuments.site/reader031/viewer/2022033021/56813679550346895d9e083a/html5/thumbnails/3.jpg)
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.
![Page 4: Area in Polar Coordinates](https://reader031.vdocuments.site/reader031/viewer/2022033021/56813679550346895d9e083a/html5/thumbnails/4.jpg)
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
![Page 5: Area in Polar Coordinates](https://reader031.vdocuments.site/reader031/viewer/2022033021/56813679550346895d9e083a/html5/thumbnails/5.jpg)
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
])([
![Page 6: Area in Polar Coordinates](https://reader031.vdocuments.site/reader031/viewer/2022033021/56813679550346895d9e083a/html5/thumbnails/6.jpg)
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
![Page 7: Area in Polar Coordinates](https://reader031.vdocuments.site/reader031/viewer/2022033021/56813679550346895d9e083a/html5/thumbnails/7.jpg)
Area of Polar Coordinates
• This all leads to the following.
![Page 8: Area in Polar Coordinates](https://reader031.vdocuments.site/reader031/viewer/2022033021/56813679550346895d9e083a/html5/thumbnails/8.jpg)
Area of Polar Coordinates
• The hardest part of this is determining the limits of integration. This is done as follows:
![Page 9: Area in Polar Coordinates](https://reader031.vdocuments.site/reader031/viewer/2022033021/56813679550346895d9e083a/html5/thumbnails/9.jpg)
Example 1
• Find the area of the region in the first quadrant that is within the cardioid r = 1 – cos.
![Page 10: Area in Polar Coordinates](https://reader031.vdocuments.site/reader031/viewer/2022033021/56813679550346895d9e083a/html5/thumbnails/10.jpg)
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
![Page 11: Area in Polar Coordinates](https://reader031.vdocuments.site/reader031/viewer/2022033021/56813679550346895d9e083a/html5/thumbnails/11.jpg)
Example 2
• Find the entire area within the cardioid r = 1 – cos.
![Page 12: Area in Polar Coordinates](https://reader031.vdocuments.site/reader031/viewer/2022033021/56813679550346895d9e083a/html5/thumbnails/12.jpg)
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
2
3)cos1(
2
0
221
2
0
221
ddrA
![Page 13: Area in Polar Coordinates](https://reader031.vdocuments.site/reader031/viewer/2022033021/56813679550346895d9e083a/html5/thumbnails/13.jpg)
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.
2
3)cos1(2
0
2
0
221
ddrA
2
3)cos1(
2
0
221
2
0
221
ddrA
![Page 14: Area in Polar Coordinates](https://reader031.vdocuments.site/reader031/viewer/2022033021/56813679550346895d9e083a/html5/thumbnails/14.jpg)
Example 3
• Find the area of the region enclosed by the rose curve r = cos2.
![Page 15: Area in Polar Coordinates](https://reader031.vdocuments.site/reader031/viewer/2022033021/56813679550346895d9e083a/html5/thumbnails/15.jpg)
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
![Page 16: Area in Polar Coordinates](https://reader031.vdocuments.site/reader031/viewer/2022033021/56813679550346895d9e083a/html5/thumbnails/16.jpg)
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.
![Page 17: Area in Polar Coordinates](https://reader031.vdocuments.site/reader031/viewer/2022033021/56813679550346895d9e083a/html5/thumbnails/17.jpg)
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
![Page 18: Area in Polar Coordinates](https://reader031.vdocuments.site/reader031/viewer/2022033021/56813679550346895d9e083a/html5/thumbnails/18.jpg)
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
![Page 19: Area in Polar Coordinates](https://reader031.vdocuments.site/reader031/viewer/2022033021/56813679550346895d9e083a/html5/thumbnails/19.jpg)
Homework
• Pages 744-745
• 1-15 odd