Q. 10 - Calculus

Question

Explain how the symmetries of the graphs of polar functions can be used to simplify area calculations.

Step-by-Step Solution

Answer

1. Substitute $\theta \mathrm{for} -\theta$. The graph is symmetric with respect to the polar axis if an equivalent equation is found.

2. Substitute $\theta \mathrm{for} -\theta \mathrm{and} r \mathrm{for} -r.$ The graph is symmetric with respect to $\theta =\frac{\pi }{2}$ if an equivalent equation is found.

3. Substitute $-r \mathrm{for} r.$ The graph is symmetric with respect to the pole if an analogous equation is found.

Step 1 : Given Information

Given : Graphs of polar functions 

Step 2 : Symmetry about the Origin

Symmetry about the origin : 

If the equation remains unchanged when $r$ is replaced by $-r$.

Example: $r^2=\cos 2\theta$

Step 3 : Symmetry about x-axis

Symmetry about $x -$ axis :

If the equation remains unchanged when $r$ is replaced by $-r$.

Example: $r=\cos 3\theta$

Step 4 : Symmetry about y-axis

Symmetry about $\underline{y-}$axis :

If the equation remains unchanged when $\theta$ is replaced by $\pi -\theta$.

Example: $r^2=\cos 2\theta$