(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?
(a) The limits of integration are -1 and 1. The derivative of f(x) is
f '(x) = -1.
The length of the curve is
(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.