10.3 day 2 Calculus of Polar Curves Lady Bird Johnson Grove, Redwood National Park, California Photo by Vickie Kelly, 2007 Greg Kelly, Hanford High School, Richland, Washington Try graphing this on the TI-89. r 2sin 2.15 0 16 To find the slope of a polar curve: dy dy d dx dx
d d r sin d d r cos d r sin r cos r cos r sin We use the product rule here. To find the slope of a polar curve: dy dy d dx dx d
d r sin d d r cos d r sin r cos r cos r sin dy r sin r cos dx r cos r sin Example: r 1 cos r sin
sin sin 1 cos cos Slope sin cos 1 cos sin 2 2 sin cos cos sin cos sin sin cos 2 2 sin cos cos 2sin cos sin cos 2 cos sin 2 sin Area Inside a Polar Graph:
The length of an arc (in a circle) is given by r. when is given in radians. For a very small , the curve could be approximated by a straight line and the area could be found using the triangle formula: A 1 bh 2 r r d 1 1 2 dA rd r r d 2 2 1 2 dA r d 2 We can use this to find the area inside a polar graph.
1 dA r 2 d 2 1 2 A r d 2 Example: Find the area enclosed by: r 2 1 cos 1 2 0 2 r d 2 1 2 4 1 cos d 0 2 2
2 2 1 2 cos cos d 2 0 2 0 1 cos 2 2 4 cos 2 d 2 2 0
1 cos 2 2 4 cos 2 d 2 2 3 4 cos cos 2 d 0 1 3 4sin sin 2 2 2 0 6 0 6 Notes:
To find the area between curves, subtract: 1 2 2 A R r d 2 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: r 2sin 2 Area of one leaf times 4: 2 1 2 A 4 2sin 2 d 2 0 A 2 Area of four leaves:
2 1 2 A 2sin 2 d 2 0 A 2 To find the length of a curve: Remember: ds dx 2 dy 2 For polar graphs: x r cos y r sin If we find derivatives and plug them into the formula, we (eventually) get: 2
dr ds r d d 2 So: Length 2 dr r d d 2
2 dr r d d 2 Length There is also a surface area equation similar to the others we are already familiar with: When rotated about the x-axis: 2 dr S 2 y r d
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