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W14 - Examples

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Polar curves

Converting to polar: pi-correction

Compute the polar coordinates of and of .

Solution

For we observe first that it lies in Quadrant II.

Next compute:

This angle is in Quadrant IV. We add to get the polar angle in Quadrant II:

The radius is of course since this point lies on the unit circle. Therefore polar coordinates are .

For we observe first that it lies in Quadrant IV. (No extra needed.)

Next compute:

So the point in polar is .

Shifted circle in polar

For example, let’s convert a shifted circle to polar. Say we have the Cartesian equation:

Then to find the polar we substitute and and simplify:

So this shifted circle is the polar graph of the polar function .

Calculus with polar curves

Finding vertical tangents to a limaçon

Let us find the vertical tangents to the limaçon (the cardioid) given by .

Convert to Cartesian parametric.

Plug into and :

Compute and .

Derivatives of both coordinates:

Simplify:

The vertical tangents occur when .

Set equation: :

Solve equation.

Convert to only :

Substitute and simplify:

Solve:

Solve for :

Compute final points.

In polar coordinates, the final points are:

In Cartesian coordinates:

For :

For :

For :

Correction: is a cusp.

The point at is on the cardioid, but the curve is not smooth there, this is a cusp.

Still, the left- and right-sided tangents exists and are equal, so in a sense we can still say the curve has vertical tangent at .

Length of the inner loop

Consider the limaçon given by . How long is its inner loop? Set up an integral for this quantity.

Solution

The inner loop is traced by the moving point when . This can be seen from the graph: 500

Therefore the length of the inner loop is given by this integral:

Area between circle and limaçon

Find the area of the region enclosed between the circle and the limaçon .

Solution

First draw the region:

200

The two curves intersect at . Therefore the area enclosed is given by integrating over :

Area of small loops

Consider the following polar graph of : 200

Find the area of the shaded region.

Solution

Bounds for one small loop.

Lower left loop occurs first.

This loop when .

Solve this:

Area integral.

Arrange and expand area integral:

Simplify integral using power-to-frequency: with :

Compute integral:

Overlap area of circles

Compute the area of the overlap between crossing circles. For concreteness, suppose one of the circles is given by and the other is given by .

Solution

Here is a drawing of the overlap:

200

Notice: total overlap area = area of red region.

Bounds: .

Apply area formula for the red region.

Area formula applied to :

Power-to-frequency: :

Double the result to include the black region: