Find an integral expression for the length of the curve f(x) = ln x on [2,4].
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.
Example 2
(a) Find an integral expression for the length of the curve f(x) = 1 – x2 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) = -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.
Example 3
(a) Write an integral expression for the length of the curve f(x) = -x on [-1,1].
(b) Evaluate your integral.
(c) Find the length of this curve without using calculus. Does your answer agree with the answer you found using the integral?
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.
