Question

Is the following statement true or false. Give an explanation and an example of your answer.

if f' is continuous on [1, 3], then the integral which goes from 1 to 3 $f' (v) d(v) = f(3) - f(1)$ ?

Answer 1

This statement is true, according to the fundamental theorem of calculus.

Let $f(x) = 2x^2$. Then $f'(x) = 4x$. Then the integral is:

$$\int_1^3 f'(x)dx = f(3)-f(1) = 2\cdot 3^2 - 2\cdot 1^2 =16$$