(a) Find an integral expression for the length of the curve *f*(*x*) = 1 – *x*^{2} on [-1,1].

(b) Use a calculator to evaluate your integral. Determine if your answer is reasonable by comparing the curve to the upper half of the unit circle.

Answer

(a) The limits of integration are *a* = -1 and *b* = 1. The derivative is

*f* '(*x*) = -2*x*.

So the length of the curve is

(b)

Here's a graph of the curve *f*(*x*) = 1 – *x*^{2} on [-1,1] and the upper half of the unit circle:

The curve *f*(*x*) = 1 – *x*^{2} falls inside the curve of the unit circle, so we would expect the length of the curve *f*(*x*) = 1 – *x*^{2} on [-1,1]

to be slightly less than half the circumference of the unit circle. Half the circumference of the unit circle is

Since 2.96 is slightly less than π, our answer is reasonable.

The limits of integration are *a* = 0 and *b* = 2π, and the derivative is

*f* '(*x*) = cos *x*.

The length of this curve is