Chapter 15.4, Problem 4CFU

To determine

To check:which of them are correct.

Answer to Problem 4CFU

Rose is correct.

Explanation of Solution

Given:

The given statement is “Cole” says thar when evaluating a definite integral, the order in which you substitute a and b in to the antiderivative and subtract does not matter.

But Rose says it does matter.

Calculation:

Rose is correct, the order does matter.

This is because ${\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)dx}=F\left(b\right)-F\left(a\right)$ .

And ${\displaystyle \underset{b}{\overset{a}{\int }}f\left(x\right)dx}=F\left(a\right)-F\left(b\right)=-\left(F\left(b\right)-F\left(a\right)\right)=-{\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)dx}$

Hence,unless the answer is 0, interchanging the order will give the opposite of the right answer.