AP Calculus BC Flashcards: The Arc Length of a Smooth, Planar Curve and Distance Traveled (BC ONLY)
Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026
Review key ideas with interactive flashcards. This set includes 10 cards to help you master important concepts.
What is Arc Length?
Arc length is the precise distance along a segment of a smooth curve in a plane. It is calculated by integrating the length of infinitesimal line segments over a given interval.
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What is Arc Length?
Arc length is the precise distance along a segment of a smooth curve in a plane. It is calculated by integrating the length of infinitesimal line segments over a given interval.
How does the concept of 'distance traveled' for a particle relate to 'arc length'?
The distance traveled by a particle along a path defined by parametric equations is equivalent to the arc length of that path over the given time interval.
Set up the definite integral to find the length of the curve y = x³ from x = 0 to x = 2.
First, find the derivative: dy/dx = 3x². The definite integral for the arc length is L = ∫[0, 2] √(1 + (3x²)²) dx.
What is a 'smooth' curve?
A curve is considered smooth on an interval if its derivatives are continuous on that interval. This condition ensures there are no sharp corners or cusps, allowing the arc length integral to be properly evaluated.
What is the definite integral formula for the arc length of a function y = f(x) from x = a to x = b?
The arc length L is calculated using the integral L = ∫[a, b] √(1 + (f'(x))²) dx.
What is the fundamental method used to determine the length of a smooth, planar curve?
The length of a planar curve defined by a function can be determined by setting up and evaluating a specific definite integral.
What is the definite integral formula for the distance traveled along a parametric curve (x(t), y(t)) from t = a to t = b?
The distance traveled, or arc length L, is calculated using the integral L = ∫[a, b] √((dx/dt)² + (dy/dt)²) dt.
What does the integrand √( (dx/dt)² + (dy/dt)² ) represent in the context of distance traveled?
The integrand represents the speed of the particle at a given time t. The definite integral of speed over a time interval gives the total distance traveled.
What is the definite integral formula for the arc length of a polar curve r = f(θ) from θ = α to θ = β?
The arc length L is calculated using the integral L = ∫[α, β] √(r² + (dr/dθ)²) dθ.
Set up the definite integral for the distance traveled from t=1 to t=4 for a particle with position (x(t) = 4t, y(t) = t²).
First, find the derivatives: dx/dt = 4 and dy/dt = 2t. The definite integral for the distance traveled is L = ∫[1, 4] √(4² + (2t)²) dt.