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-84.
2sin 2.15
0 16
r
2 2 2cos r
ysin tan =
x
x r x y
y r
8sinr 6cos 6sinr
Polar-Rectangular Conversion Formulas
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
• Find the slope of the curve at the given values.
• Find the points where the curve has horizonal or vertical tangent lines.
8sin 3r / 2, 2 / 3 1 sinr
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.
Find the area of the region that lies inside the circle r = 3 and outside the cardioid . 3 1 cosr
x
y
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
• Find the area that lies outside the four-petal rose and inside the circle.
3cos 2
3
r
r
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
Or…
• Convert to Parametric!
• Find the length of the cardioid
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