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