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Photo by Vickie Kelly, 2007. Greg Kelly, Hanford High School, Richland, Washington. 10.3 day 2 Calculus of Polar Curves. Lady Bird Johnson Grove, Redwood National Park, California. Try graphing this on the TI-89. To find the slope of a polar curve:. We use the product rule here. - PowerPoint PPT Presentation
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10.3 day 2Calculus of Polar Curves
Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2007
Lady Bird Johnson Grove,Redwood National Park, California
Try graphing this on the TI-89.
2sin 2.15
0 16
r
To find the slope of a polar curve:
dy
dy ddxdxd
sin
cos
dr
ddr
d
sin cos
cos sin
r r
r r
We use the product rule here.
To find the slope of a polar curve:
dy
dy ddxdxd
sin
cos
dr
ddr
d
sin cos
cos sin
r r
r r
sin cos
cos sin
dy r r
dx r r
Example: 1 cosr sinr
sin sin 1 cos cosSlope
sin cos 1 cos sin
2 2sin cos cos
sin cos sin sin cos
2 2sin cos cos
2sin cos sin
cos 2 cos
sin 2 sin
The length of an arc (in a circle) is given by r. when is given in radians.
Area Inside a Polar Graph:
For a very small , the curve could be approximated by a straight line and the area could be found using the triangle formula: 1
2A bh
r dr
21 1
2 2dA rd r r d
We can use this to find the area inside a polar graph.
21
2dA r d
21
2dA r d
21
2A r d
Example: Find the area enclosed by: 2 1 cosr
2 2
0
1
2r d
2 2
0
14 1 cos
2d
2 2
02 1 2cos cos d
2
0
1 cos 22 4cos 2
2d
2
0
1 cos 22 4cos 2
2d
2
03 4cos cos 2 d
2
0
13 4sin sin 2
2
6 0
6
Notes:
To find the area between curves, subtract:
2 21
2A R r d
Just like finding the areas between Cartesian curves, establish limits of integration where the curves cross.
When finding area, negative values of r cancel out:
2sin 2r
22
0
14 2sin 2
2A d
Area of one leaf times 4:
2A
Area of four leaves:
2 2
0
12sin 2
2A d
2A
To find the length of a curve:
Remember: 2 2ds dx dy
For polar graphs: cos sinx r y r
If we find derivatives and plug them into the formula, we (eventually) get:
22 dr
ds r dd
So: 22Length
drr d
d
22Length
drr d
d
There is also a surface area equation similar to the others we are already familiar with:
22S 2
dry r d
d
When rotated about the x-axis:
22S 2 sin
drr r d
d