Problem

Find the length of the parabola y = x² from x = 0 to x = 1.

Solution

The arclength formula is

Here, a = 0, b = 1, and

The integral is not one we know how to solve as yet, so we will estimate it using the trapezoid rule with the Numerical Integration applet.

Choose a value for n.

This is the value of the estimate, shown to 3 decimal places.

There is an exact antiderivative, that we can find by computer or a lookup table.

asinh is the inverse hyperbolic sin function. The exact value of the integral to 3 decimal places is