Polar Area Between Curves — Theory
1. Area in polar coordinates
A polar curve is written as
\[
r=f(\theta).
\]
The area swept out from \(\theta=a\) to \(\theta=b\) is
\[
A=\frac12\int_a^b r^2\,d\theta.
\]
The factor \(\frac12\) comes from the area of a thin circular sector.
2. Area between two polar curves
For two polar curves
\[
r_1=f(\theta),
\qquad
r_2=g(\theta),
\]
the area between them is
\[
A=\frac12\int_a^b
\left|r_1^2-r_2^2\right|
\,d\theta.
\]
The absolute value is useful when the outer curve changes inside the interval.
3. Outside curve minus inside curve
If one curve is known to be outside the other for the whole interval, then the formula becomes
\[
A=\frac12\int_a^b
\left(r_{\text{outer}}^2-r_{\text{inner}}^2\right)
\,d\theta.
\]
This is often the simplest form when the curves do not cross inside the interval.
4. Why squared radii appear
A small sector with radius \(r\) and angle width \(d\theta\) has area
\[
dA=\frac12r^2\,d\theta.
\]
Therefore, the area between two polar curves depends on \(r_1^2-r_2^2\), not simply on \(r_1-r_2\).
5. Intersection points
Intersections help decide where one curve becomes outside the other. A practical polar-area check is
\[
r_1(\theta)^2=r_2(\theta)^2.
\]
This catches both \(r_1=r_2\) and \(r_1=-r_2\), which matters because negative polar radii can represent
points across the pole.
6. Example: circle and rose curve
Suppose
\[
r_1=2,
\qquad
r_2=3\sin\theta.
\]
On a chosen interval, the area between the curves is
\[
A=\frac12\int_a^b
\left|2^2-\left(3\sin\theta\right)^2\right|
\,d\theta.
\]
Intersections occur where
\[
4=9\sin^2\theta.
\]
7. Converting polar points to graph points
The graph is drawn in the Cartesian plane using
\[
x=r\cos\theta,
\qquad
y=r\sin\theta.
\]
This lets the shaded polar region be displayed on an ordinary \(x\)-\(y\) graph.
8. Units on graph axes
If radius is measured in meters, then both Cartesian coordinates are also measured in meters:
\[
x\;(\mathrm{m}),
\qquad
y\;(\mathrm{m}).
\]
Area is then measured in square meters:
\[
A\;(\mathrm{m}^2).
\]
The calculator includes a distance-unit field and automatically reports area in the squared unit.
9. Numerical integration
Many polar-curve area problems cannot be integrated easily by hand. Numerical integration estimates the area
by sampling the integrand
\[
\frac12\left|r_1^2-r_2^2\right|.
\]
This calculator compares trapezoid and Simpson estimates.
10. Simpson rule
Simpson rule is often accurate for smooth functions. It combines sampled values in the pattern
\[
A\approx
\frac{\Delta\theta}{3}
\left[
f_0+4f_1+2f_2+4f_3+\cdots+f_n
\right].
\]
It requires an even number of subintervals and finite sampled values.
11. Trapezoid rule
The trapezoid rule uses straight-line segments between sampled values:
\[
A\approx
\frac{\Delta\theta}{2}
\left[
f_0+2f_1+2f_2+\cdots+f_n
\right].
\]
It is also useful as a fallback when some samples are invalid or the curves behave sharply.
13. Common mistakes
- Forgetting the factor \(\frac12\): polar area uses \(\frac12r^2\), not just \(r^2\).
- Subtracting radii instead of squared radii: use \(r_1^2-r_2^2\), not \(r_1-r_2\).
- Ignoring intersections: if curves cross, the outside curve may change.
- Using degrees accidentally: polar integrals use radians.
- Ignoring negative radii: negative \(r\) can place points across the pole.
- Dropping units: graph axes use distance units, and area uses squared units.
