Stewart Calculus · 8th Edition
Chapter 10: Parametric Equations & Polar Coordinates
Every key formula from Stewart Calculus Chapter 10, in one searchable page. Click a card to study it — worked examples included.
10.1 · PARAMETRIC CURVES
x = f(t), y = g(t)
Eliminate t to get y(x) when possible.
Same curve, different t = different speed/direction.
Same curve, different t = different speed/direction.
Cycloid
x = r(θ − sin θ), y = r(1 − cos θ)
Path of a point on a rolling wheel of radius r.
Parametric line
x = x₀ + at, y = y₀ + bt
Through (x₀, y₀) in direction ⟨a, b⟩.
WORKED EXAMPLE
x = 1 + 3t, y = 2 − t²: t = (x−1)/3 ⇒
y = 2 − (x − 1)²/9 (parabola).
y = 2 − (x − 1)²/9 (parabola).
Eliminate t: solve one equation for t, substitute into the other.
dy/dx = dy/dtdx/dt, d²y/dx² = d/dt(dy/dx)dx/dt
L = ∫αβ √((x′)² + (y′)²) dt
WORKED EXAMPLE
x = t², y = t³ − t at t = 1:
dy/dx = (3t² − 1)/(2t) = 1; point (1, 0).
dy/dx = (3t² − 1)/(2t) = 1; point (1, 0).
Tangents
Horizontal: dy/dt = 0; vertical: dx/dt = 0; cusp: both 0.
Area under the curve
A = ∫αβ g(t) f′(t) dt
Trace left → right so A > 0.
WORKED EXAMPLE
One arch of the cycloid, 0 ≤ θ ≤ 2π:
L = ∫02π r√(2 − 2cos θ) dθ = 8r.
L = ∫02π r√(2 − 2cos θ) dθ = 8r.
10.3 · POLAR COORDINATES
x = r cos θ, y = r sin θ
r² = x² + y², tan θ = y/x
WORKED EXAMPLE
r = 2 sin θ ⇒ x² + y² = 2y ⇒
x² + (y − 1)² = 1 (circle).
x² + (y − 1)² = 1 (circle).
Symmetry tests
r(−θ) = r(θ) ⇒ x-axis; r(π−θ) = r(θ) ⇒ y-axis; r(θ+π) = r(θ) ⇒ pole.
Gallery
r = a circle; θ = α line through pole;
r = 2a sin θ circle through pole; r = a ± b cos θ limaçon; r = a cos nθ rose.
r = 2a sin θ circle through pole; r = a ± b cos θ limaçon; r = a cos nθ rose.
KEY NOTES
- Polar coordinates are not unique: (r, θ) = (r, θ+2π) = (−r, θ+π).
10.4 · AREAS IN POLAR
A = ∫αβ ½ r² dθ
Find the θ-range for ONE loop first.
WORKED EXAMPLE
Cardioid r = 1 + cos θ:
A = ½∫02π (1 + cos θ)² dθ = 3π2.
A = ½∫02π (1 + cos θ)² dθ = 3π2.
Between curves
A = ∫αβ ½(R² − r²) dθ
Outer R minus inner r.
WORKED EXAMPLE
One petal of r = sin 2θ (0 ≤ θ ≤ π/2):
A = ½∫0π/2 sin²2θ dθ = π/8.
A = ½∫0π/2 sin²2θ dθ = π/8.
WORKED EXAMPLE
Inside r = 2 + 2cos θ, outside r = 2:
meet at θ = ±π/2; A = ∫−π/2π/2 ½[(2+2cos θ)² − 4] dθ = 8 + π.
meet at θ = ±π/2; A = ∫−π/2π/2 ½[(2+2cos θ)² − 4] dθ = 8 + π.
10.5–10.6 · LENGTHS & CONICS
L = ∫αβ √(r² + (dr/dθ)²) dθ
Conics in polar
r = ed1 ± e cos θ
e < 1 ellipse, e = 1 parabola, e > 1 hyperbola.
Cartesian forms
x²/a² + y²/b² = 1 ellipse; y² = 4px parabola;
x²/a² − y²/b² = 1 hyperbola.
x²/a² − y²/b² = 1 hyperbola.
WORKED EXAMPLE
Perimeter of cardioid r = 1 + cos θ:
L = ∫02π √(2 + 2cos θ) dθ = 8.
L = ∫02π √(2 + 2cos θ) dθ = 8.
Cardioid r = a(1 + cos θ): cusp at the pole, axis along θ = 0.
WORKED EXAMPLE
r = 4/(2 + cos θ) = 2/(1 + ½cos θ):
e = 1/2 < 1 ⇒ ellipse; vertices r = 4/3 (θ = 0), r = 4 (θ = π).
e = 1/2 < 1 ⇒ ellipse; vertices r = 4/3 (θ = 0), r = 4 (θ = π).
WATCH OUT!
Mistakes that cost points
- d²y/dx² ≠ (d²y/dt²)/(d²x/dt²) — differentiate dy/dx, divide by x′
- Polar area: the ½ is essential
- tan θ = y/x needs a quadrant check
- r = a cos(nθ): n odd ⇒ n petals, n even ⇒ 2n petals
- θ-limits must trace the curve ONCE — a repeated loop double-counts area
- r(π−θ) = r(θ) is y-axis symmetry, not x-axis
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