Section 9.10

Surface Area in Polar

When we rotate a polar curve, we generate a surface. To measure it, we sum up an infinite number of thin ribbons or "bands".

The Band Method

Imagine wrapping a ribbon around the shape. The area of one tiny ribbon is:
Circumference × Width

Circumference (): The distance around the rotation. If rotating about the Polar Axis (x-axis), the radius is the vertical height .
Width (): The slant length of the curve itself. As we learned, .

Putting it together for rotation about the Polar Axis:

Rotate circle about the polar axis.

1. Arc Element (ds)

2. Integral

Radius to axis is .

(Only integrate to for top half!)

3. Result

.
Matches (with ).

Rotating the yellow arc about the dashed axis.

Rotate about the polar axis.

1. Setup

(from Arc Length example).
Radius arm = .

2. Integrate

Bounds: to (Top half).

Result:

Rotate about the polar axis.

1. Identify the Generator

The curve exists in two loops.
Right Loop: to .

WARNING: Do not integrate

The bottom part ( to ) is the mirror image of the top. Rotating it would count the same surface twice.

Use to .

2. Symmetry

This gives the RIGHT bulb. By symmetry, multiply by 2 for the LEFT bulb.
.

Only rotate the yellow/top section to generate the surface.