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  2. AP Calculus
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Arc length formula for parametric curves?

S=∫ab(dx(t)dt)2+(dy(t)dt)2dtS=\int_a^b \sqrt{(\tfrac{dx(t)}{dt})^2 + (\tfrac{dy(t)}{dt})^2} dtS=∫ab​(dtdx(t)​)2+(dtdy(t)​)2​dt

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Arc length formula for parametric curves?

S=∫ab(dx(t)dt)2+(dy(t)dt)2dtS=\int_a^b \sqrt{(\tfrac{dx(t)}{dt})^2 + (\tfrac{dy(t)}{dt})^2} dtS=∫ab​(dtdx(t)​)2+(dtdy(t)​)2​dt

Arc length formula for Cartesian curves?

S=∫ab1+[f′(x)]2dxS=\int_a^b \sqrt{1+[f'(x)]^2} dxS=∫ab​1+[f′(x)]2​dx

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.