Finding Arc Length of y = x√x

Calculus 1 · Axiom Academy

Measuring the true length of a curve with the arc length integral and a u-substitution. Find the arc length of the curve y = x^ 3/2 from x = 0 to x = 4 . The curve y = x^ 3/2 on [0, 4] . Arc length is the length of this bold path from (0,0) to (4,8) — not the straight line, and not the area beneath it. Nice work — you measured a curved length end to end by turning it into a single definite integral. The formula: — the only setup you need. Why it simplified: squaring kills the square root, leaving the linear under the radical. The u-substitution: makes the integral a routine power rule; the limits ride along from u=1 to u=10 . Result: , a touch longer than the straight-line distance . The " +1 under the root" is the whole trick: it bakes the run and the rise into one length, so a hard-looking curve becomes an integral you can actually evaluate.

This is the written version of the interactive lesson above. See the full Calculus 1 course.