10.3 day 2 calculus of polar curves greg kelly, hanford high school, richland, washingtonphoto by...
TRANSCRIPT
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