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How does arc length relate to the Pythagorean Theorem?
Arc length is approximated by summing the hypotenuses of small right triangles with sides dx and dy.
Why do we need a different arc length formula for parametric curves?
Because both x and y coordinates are changing with respect to a parameter t, so we must account for both rates of change.
Define arc length.
The distance between two points along a curve.
What is a parametric curve?
A curve where the x and y coordinates are defined by functions of a parameter, usually denoted as t.
Define a smooth planar curve.
A curve in a plane that has a continuous first derivative.
Steps to find arc length of parametric curve?
- Find dx/dt and dy/dt. 2. Substitute into the arc length formula. 3. Evaluate the integral.
How to evaluate ?
- Integrate to get [t]. 2. Evaluate at the bounds: .