Length of y = x√x from 0 to 4

Calculus 2 · Axiom Academy

Finding the length of a curve from x = 0 to x = 4 using the arc length formula and u-substitution. Find the arc length of the curve y = x^ 3/2 from x = 0 to x = 4 . Excellent work! You've successfully computed an arc length integral. Here's what we learned: Arc Length Formula: For y = f(x) , the arc length from a to b is . Derivative First: Always find and square the derivative before setting up the integral. Simplify the Radicand: Combine terms under the square root carefully — here we got . U-Substitution: Arc length integrals often require substitution to evaluate — let u equal the expression inside the square root. Change Limits: When using u-substitution with definite integrals, convert the x -limits to u -limits. Why Arc Length is Hard: The square root of a sum (not a product) creates integrals that rarely have simple antiderivatives — many require numerical methods. This problem demonstrates a relatively "nice" arc length integral that can be solved exactly. Most arc length problems in practice require numerical integration techniques like Simpson's Rule or technology to approximate the answer.

This is the written version of the interactive lesson above. See the full Calculus 2 course.