Polar Coordinates
& Calculus
Describe a curve with a turning ray.
Use that picture to find slopes, areas, and lengths.
Turn through θ, then move by r
The angle chooses a direction; the sign of the radius chooses a side.
Polar coordinates $(r,\theta)$
- The pole is the origin. The polar axis is the positive $x$-axis.
- Positive $\theta$ turns counterclockwise; angles are in radians.
- $r>0$: move along the ray. $r<0$: move in the opposite direction.
- The distance to the pole is $|r|$.
One point, many names
Every $(0,\theta)$ represents the pole.
The same point, two coordinate systems
Right-triangle projections give the conversion formulas.
Polar → Cartesian
Example: $(2,\pi/3)\longmapsto(1,\sqrt3)$.
Cartesian → polar (choose $r\geq0$)
Choose $\theta$ in the correct quadrant. On the axes, use the direction directly.
Vertical projection: $r\sin\theta$.
The projection identities hold for every angle and for either sign of $r$.
Simple polar equations describe familiar curves
Convert with $x=r\cos\theta$, $y=r\sin\theta$, and $r^2=x^2+y^2$.
A constant radius
Circle centered at the pole, radius $2$.
A constant angle
Allowing all real $r$ gives the whole line. Restricting $r\geq0$ gives a ray.
A shifted circle
$x^2+y^2=2x$, so
$(x-1)^2+y^2=1$.
The pole is on both curves; check it when multiplying by $r$.
Read r(θ), then trace the point in the plane
The left graph gives the radius; the right graph plots $(r\cos\theta,r\sin\theta)$.
Recognize the shape from the radius rule
For the formulas below, $a>0$ and $n$ is a positive integer.
Cardioid
Maximum radius $2a$.
Rose
$n$ even: $2n$ petals.
Petal length $a$.
Lemniscate
$\cos(2\theta)\geq0$.
Spiral
The tracing interval matters
$r=\cos(3\theta)$ traces its three petals once on $[0,\pi]$; $[0,2\pi]$ traces them twice. By comparison, $r=\cos(2\theta)$ needs $[0,2\pi]$ for all four petals.
Treat θ as a parameter to find a tangent
The radius changes while the ray rotates: both motions affect the slope.
Write $r'=dr/d\theta$. The product rule gives
Horizontal: $y'=0$ and $x'\ne0$.
Vertical: $x'=0$ and $y'\ne0$.
If both vanish, examine the limiting tangent. At the pole, if $r'\ne0$, the tangent is the line in direction $\theta$.
tangent radius
At $\theta=\pi/2$: $(x,y)=(0,\pi/2)$ and $m=-2/\pi$.
Build area from thin circular sectors
A small angle sweeps a narrow wedge; add the wedge areas.
Why the factor $\tfrac12 r^2$?
Freeze the radius in each small wedge. As the wedges get thinner, the sum approaches the region's area.
Solid blue: exact boundary. Filled wedges: midpoint approximation. Angles must be in radians.
Find one petal by locating consecutive zeros
Find the area of the right petal of $r=\cos(2\theta)$.
1. Identify the interval
$\cos(2\theta)=0$ at $\theta=-\pi/4,\pi/4$. Between them, $r\geq0$ and the right petal is traced once.
2. Square the radius and integrate
All four petals: $4(\pi/8)=\pi/2$.
Subtract squared radii on the same ray
Find the area inside $r=3\sin\theta$ and outside $r=1+\sin\theta$.
Use $R\geq r\geq0$; split wherever the outer curve changes.
Find the bounding rays
$3\sin\theta=1+\sin\theta$ gives $\theta=\pi/6,\,5\pi/6$.
Show evaluation (use symmetry)
$R=3\sin\theta$ $r=1+\sin\theta$
Outer sector minus inner sector: $R^2-r^2$, not $(R-r)^2$.
Length combines radial motion and turning
A tiny displacement has two perpendicular components.
Radial change: $dr$ along the ray.
Turning: $r\,d\theta$ perpendicular to it.
Assume $r$ is continuously differentiable (or split into smooth pieces), and trace the desired curve once.
Arrows share a scale: radial $r'$, turning $r$, total velocity per radian.
A cardioid has an exact length of 8
Use the rotated cardioid $r=1+\cos\theta$, traced once on $[-\pi,\pi]$.
Simplify before integrating
On $[-\pi,\pi]$, $\cos(\theta/2)\geq0$, so
Scaling every radius by $a$ scales every length by $a$.
Exercise: Four-Petaled Rose and Circle
Handle multiloop intersections, double-angle identities, and boundary arc lengths.
1 · Petal Intersections & Interval
Find the intersection points of the rose curve $r=2\cos(2\theta)$ and the circle $r=1$ on the right petal. Which $\theta$-interval isolates the tip region of this petal outside the circle?
Reveal detailed solution
1. Solve for $\theta$: $2\cos(2\theta)=1 \implies \cos(2\theta)=\frac12 \implies 2\theta=\pm\frac\pi3 \implies \theta=\pm\frac\pi6$.
2. Coordinates: Polar pairs are $(1,\frac\pi6)$ and $(1,-\frac\pi6)$. In Cartesian coordinates: $x=1\cos(\frac\pi6)=\frac{\sqrt3}{2}$, $y=1\sin(\pm\frac\pi6)=\pm\frac12$.
3. Sweep interval: On $[-\frac\pi6,\frac\pi6]$, $r_{\text{rose}}=2\cos(2\theta)\ge r_{\text{circle}}=1$.
2 · Exact Area of the Petal Tip
Find the exact area of the region lying inside the right petal of $r=2\cos(2\theta)$ and outside the circle $r=1$.
Reveal detailed solution
Apply $\cos^22\theta=\frac{1+\cos4\theta}{2}$, so $4\cos^22\theta-1=1+2\cos4\theta$:
3 · Exact Perimeter of R
Find the exact perimeter of $R$. Your answer should include the appropriate boundary arcs from both curves. (You do not need to evaluate the outer arc integral; setting up the formula is enough.)
Reveal detailed solution
1. Inner circular arc ($r=1$): On $r=1$, $ds=1\,d\theta$:
2. Outer rose arc ($r=2\cos 2\theta$): $r'=-4\sin 2\theta \implies ds=\sqrt{r^2+(r')^2}\,d\theta$:
3. Total perimeter:
Note: The outer integral is an elliptic integral with no elementary closed form, so writing the exact definite integral formula is ok.
Rose $r=2\cos(2\theta)$ Circle $r=1$
Shaded: petal tip region. Radial segment $R(\theta)-1$.
Keep the geometry attached to each formula
Locate the point → trace the curve → choose the appropriate calculus formula.
Coordinates · projections
Check the quadrant and the sign of $r$.
Tangent · ratio of velocities
Use when the denominator is nonzero. Check horizontal, vertical, and singular cases separately.
Area · sum of sectors
Sweep once; for subtraction use $R\geq r\geq0$ on the same ray.
Length · radial + turning motion
Use smooth pieces and trace each part once.
Remember $\sqrt{u^2}=|u|$.
Based on Stewart, Calculus: Early Transcendentals, 9e, §§10.3–10.4, pp. 684–699. Textbook excerpt · Instructor’s guide · Concept check. Diagrams are interactive illustrations of the formulas.