A student computes the arc length of r = 5 (a circle of radius 5) from θ = 0 to θ = 2π using L = ∫₀^{2π} |dr/dθ| dθ. What result do they get, and why is it wrong?
AThey get L = 0, because dr/dθ = 0 for a constant radius — the formula misses the angular component of motion entirely.
BThey get L = 10π, which is the correct circumference of the circle.
CThey get L = 5, because the radius is 5 regardless of θ.
DThey get L = 2π, from integrating 1 dθ without the radius factor.
Since r = 5 is constant, dr/dθ = 0 everywhere, so ∫|dr/dθ| dθ = 0. But a point sweeping around a circle of radius 5 clearly travels a distance of 10π — the circumference. The error is treating arc length as though only radial motion (change in r) contributes. In polar coordinates, angular motion also covers distance: even with constant r, moving through angle dθ covers a distance of r dθ. The r² term in the correct formula √(r² + (dr/dθ)²) captures this angular component.
Question 2 Multiple Choice
Which feature of the polar arc length formula L = ∫√(r² + (dr/dθ)²) dθ most distinguishes it from the naive formula L = ∫|dr/dθ| dθ?
AThe polar formula uses θ as the integration variable rather than x.
BThe polar formula is derived from the Pythagorean theorem, while the naive formula is not.
CEven when dr/dθ = 0 (constant radius), the polar formula correctly yields nonzero arc length because angular motion contributes actual distance.
DThe polar formula involves a square root, which accounts for the curvature of the coordinate system.
The r² term captures the distance covered by angular motion. When r is constant, dr/dθ = 0, and the formula gives ∫√(r²) dθ = ∫r dθ = r·Δθ — the standard arc length for a circular arc. The naive formula gives zero in this case, which is clearly wrong. The key insight is that polar motion has two components: radial (dr/dθ) and angular (r), and the formula accounts for both.
Question 3 True / False
The polar arc length formula L = ∫√(r² + (dr/dθ)²) dθ is derived by treating the polar curve as a parametric curve and substituting x = r cos θ, y = r sin θ into the parametric arc length formula.
TTrue
FFalse
Answer: True
This is exactly how the formula is derived. Setting x(θ) = r cos θ and y(θ) = r sin θ, computing dx/dθ and dy/dθ, then squaring and adding yields (dx/dθ)² + (dy/dθ)² = (dr/dθ)² + r². The polar arc length formula follows directly from the parametric formula L = ∫√((dx/dθ)² + (dy/dθ)²) dθ.
Question 4 True / False
The polar arc length formula A = ½∫r² dθ and the arc length formula L = ∫√(r² + (dr/dθ)²) dθ compute different quantities, but are equivalent when dr/dθ is small.
TTrue
FFalse
Answer: False
These formulas compute fundamentally different things — area and arc length — and no assumption about dr/dθ makes them equivalent. The area formula accumulates infinitesimal sector slices (proportional to r²); the arc length formula accumulates infinitesimal distances (involving r² and (dr/dθ)²). The similarity in notation (both integrate in θ, both involve r²) is superficial. Always identify which quantity is being requested before setting up the integral.
Question 5 Short Answer
Why does the polar arc length formula include an r² term inside the square root, even for a curve where dr/dθ = 0 throughout?
Think about your answer, then reveal below.
Model answer: Because even when the radius isn't changing, a point moving along the curve is still sweeping through angle θ, covering actual distance proportional to r per radian. The r² term captures this angular component of motion. Without it, a circle (constant r, dr/dθ = 0 everywhere) would be computed to have zero arc length — clearly wrong. The formula accounts for both radial motion (dr/dθ) and angular motion (r) simultaneously.
This is the central insight of the polar arc length formula. Motion in polar coordinates is inherently two-dimensional: you can move radially (changing r) or angularly (sweeping θ), and both contribute to arc length. The parametric derivation makes this explicit: (dx/dθ)² + (dy/dθ)² = (dr/dθ)² + r², so both components always appear.