Chapter 6.2, Problem 25E

To determine

Toevaluate

  ${\displaystyle \underset{a}{\overset{b}{\int }}2sds}$

Answer to Problem 25E

  ${\displaystyle \underset{a}{\overset{b}{\int }}2sds}=b^2-a^2$

Explanation of Solution

Given:

  ${\displaystyle \underset{a}{\overset{b}{\int }}2sds};0<a<b$

Calculation: Consider the integral

  $\begin{array}{l}I={\displaystyle \underset{a}{\overset{b}{\int }}2sds}\\ =2{\displaystyle \underset{a}{\overset{b}{\int }}sds}\\ =2{\left[\frac{s^2}{2}\right]}_a^b\\ =2\left[\frac{b^2}{2}-\frac{a^2}{2}\right]\\ =2\left[\frac{b^2-a^2}{2}\right]\\ =b^2-a^2\end{array}$