Here a small piece of the curve is equal approximately to .
As Δx → 0, the sum of these pieces approaches the definite integral above.
If the derivative of the function x = g(y) is continuous on the interval c ≤ y ≤ d,
then the length s of the arc from y = c to y = d is given by
If a curve is defined parametrically by the equations x = x(t) and y = y(t), if the
derivatives of the functions x(t) and y(t) are continuous on [ta , tb] (and if the
curve does not intersect itself), then the length of the arc from t = ta to t = tb is
given by
BC ONLYThe 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 N7–21Example 13 __
Find the length, to three decimal places, of the arc of y = x3/2 from x = 1 to x = 8.
SOLUTION: Here .