Area, Volume, and Arc Length Exercises

Answer

The limits of integration are a = 2 and b = 4. The derivative is

The length of a little bit of curve is

and the length of the curve is

There's a formula for everything here. We just need to find f '(x) and then plug everything in.

Answer

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

f '(x) = -2x.

So the length of the curve is

(b) 

Here's a graph of the curve f(x) = 1 – x² on [-1,1] and the upper half of the unit circle:

The curve f(x) = 1 – x² falls below the curve of the upper half of the unit circle, so we would expect the length of the curve f(x) = 1 – x² on [-1,1] to be a little 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.

Answer

(a) The limits of integration are -1 and 1. The derivative of f(x) is

f '(x) = -1.

The length of the curve is

(b)

(c) This curve is just a straight line.

The change in x on this interval is 2 and the change in y is -2.

Using the Pythagorean Theorem, the length of this line is

This agrees with the answer we found using the fancy formula.