Calculus with polar curves
05 Theory - Polar tangent lines, arclength
Theory 1
Polar arclength formula
The arclength of the polar graph of
, for : To derive this formula, convert to Cartesian with parameter
: From here you can apply the familiar arclength formula with
in the place of . Link to originalExtra - Derivation of polar arclength formula
Let
and convert to parametric Cartesian, so: Then:
Therefore:
Therefore:
06 Illustration
Example - Length of the inner loop
Length of the inner loop
Consider the limaçon given by
. How long is the 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:
Therefore the length of the inner loop is given by this integral:
Link to original
07 Theory - Polar area
Theory 2
Sectorial area from polar curve
The “area under the curve” concept for graphs of functions in Cartesian coordinates translates to a “sectorial area” concept for polar graphs. To compute this area using an integral, we divide the region into Riemann sums of small sector slices.
To obtain a formula for the whole area, we need a formula for the area of each sector slice.
Area of sector slice
Let us verify that the area of a sector slice is
.
Take the angle
in radians and divide by to get the fraction of the whole disk. Then multiply this fraction by
(whole disk area) to get the area of the sector slice. Now use
and for an infinitesimal sector slice, and integrate these to get the total area formula:
One easily verifies this formula for a circle.
Let
be a constant. Then:
The sectorial area between curves:
Sectorial area between polar curves
Subtract after squaring, not before!
This aspect is not similar to the Cartesian version:
08 Illustration
Area between circle and limaçon
Area between circle and limaçon
Find the area of the region enclosed between the circle
and the limaçon . Solution
First draw the region:
The two curves intersect at
. Therefore the area enclosed is given by integrating over : Link to original
Area of small loops
Area of small loops
Consider the following polar graph of
:
Find the area of the shaded region.
Solution
Find bounds for one small loop. Lower left loop occurs first. This loop is when
. Now set up area integral:
Power-to-frequency conversion:
with : Link to original
Overlap area of circles
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
Drawing of the overlap:
Notice: total overlap area =
area of red region. Bounds for red region: . Area formula applied to
: Power-to-frequency:
: Link to original
Complex algebra
Videos
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01 Theory - Complex arithmetic
Theory 1
The complex numbers
are sums of real and imaginary numbers. Every complex number can be written uniquely in ‘Cartesian’ form: To add, subtract, scale, and multiply complex numbers, treat ‘
’ like a constant. Simplify the result using
. For example:
Complex conjugate
Every complex number has a complex conjugate:
For example:
In general,
. Conjugates are useful mainly because they eliminate imaginary parts:
In general:
Complex division
To divide complex numbers, use the conjugate to eliminate the imaginary part in the denominator.
For example, reciprocals:
More general fractions:
Multiplication preserves conjugation
For any
: Therefore, one can take products or conjugates in either order.
02 Illustration
Example - Complex multiplication
Complex multiplication
Compute the products:
(a)
(b) Solution
(a)
Expand:
Simplify
:
(b)
Expand:
Simplify
: Link to original
Example - Complex division
Complex division
Compute the following divisions of complex numbers:
(a)
(b) (c) (d) Solution
(a)
Conjugate is
: Simplify:
(b)
Conjugate is
:
(c)
Factor out the
: Use
:
(d)
Denominator conjugate is
: Simplify:
Link to original