Barrons AP Calculus - David Bock

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The parenthetical clause above is equivalent to the requirement that the curve is traced out just once
as t varies from ta to tb.
As indicated in Equation (4), formulas (1), (2), and (3) can all be derived easily from the very
simple relation

and can be remembered by visualizing Figure N7–21.

FIGURE N7–21

EXAMPLE 13

Find the length, to three decimal places, of the arc of y = x3/2 from x = 1 to x = 8.

SOLUTION: Here

EXAMPLE 14

Find the length, to three decimal places, of the curve (x − 2)^2 = 4y^3 from y = 0 to y = 1.

SOLUTION: Since x − 2 = 2y3/2 and Equation (2) above yields

EXAMPLE 15

The position (x, y) of a particle at time t is given parametrically by x = t^2 and Find the

distance the particle travels between t = 1 and t = 2.

SOLUTION: We can use (4): ds^2 = dx^2 + dy^2 , where dx = 2 t dt and dy = (t^2 − 1) dt. Thus,

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