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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} 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} dt

Arc length formula for Cartesian curves?

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

Steps to find arc length of parametric curve?

  1. Find dx/dt and dy/dt. 2. Substitute into the arc length formula. 3. Evaluate the integral.

How to evaluate ∫0π1dt\int_0^\pi \sqrt{1} dt?

  1. Integrate to get [t]. 2. Evaluate at the bounds: π−0=π\pi - 0 = \pi.

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.