Arc Length
Definition
The arc length of a curve is the distance along the curve between two points. It measures how long the curve is.
Analogy
Imagine you are walking along a curvy path in a park. The arc length is like the total distance you walk from the starting point to the ending point, following every twist and turn of the path.
Related terms
Parametric Equations: Equations that express variables in terms of parameters. They are often used to describe curves or paths in space.
Tangent Line: A line that touches a curve at only one point and has the same slope as the curve at that point.
Derivative: The rate at which a function changes with respect to its input variable. It represents the slope of a tangent line to a curve at any given point.
"Arc Length" appears in:
Study guides (2)
Practice Questions (20+)
- Which of the following formulas calculates the arc length of a smooth, planar curve?
- What is the arc length of the curve y = x^2, from x = 0 to x = 2?
- The arc length of a curve is always _______ the distance traveled along the curve.
- Which of the following integrals represents the arc length of the curve y = e^x, from x = 0 to x = 1?
- What is the arc length of the curve y = sin(x), from x = 0 to x = π?
- If the equation of a curve is given parametrically by x = t^2 and y = t^3, what is the arc length of the curve from t = 0 to t = 1?
- The arc length of a curve is always a _______ quantity.
- The distance traveled by an object is always _______ the arc length of the curve representing its motion.
- The arc length of a curve is always _______ the displacement of an object along the curve.
- Which of the following is an application of arc length in real-world scenarios?
- The arc length of a curve can be approximated by dividing the curve into _______.
- Which of the following integrals represents the arc length of the curve y = ln(x), from x = 1 to x = e?
- What is the formula to find arc length for a parametric equation?
- Consider the parametric equations: x(t) = t^2 y(t) = 3t What is the arc length of the curve between t = 0 and t = 2?
- The parametric equations of a curve are: x(t) = 2cos(t) y(t) = 2sin(t) What is the arc length of the curve between t = 0 and t = π/2?
- Consider the parametric equations: x(t) = e^t + e^(-t) y(t) = e^t - e^(-t) What is the arc length of the curve between t = 0 and t = ln(2)?
- Consider the parametric equations: x(t) = t^2 - 1 y(t) = t^3 + 2t What is the arc length of the curve between t = -1 and t = 1?
- Consider the curve given by the equation y = x^2. What is the arc length of the curve between x = 0 and x = 1?
- The curve y = sin(x) is defined over the interval [0, π/2]. What is the arc length of the curve?
- For the equation y = ln(x), what is the arc length of the curve between x = 1 and x = e?
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