Chapter 6.2, Problem 6E

To determine

To express the given limit as a definite integral.

Answer to Problem 6E

  ${\displaystyle \underset{-\pi }{\overset{\pi }{\int }}{\sin }^{3}xdx}$

Explanation of Solution

Given:

  $\underset{n\to \infty }{\lim }{\displaystyle \sum _{k=1}^n\left({\sin }^3{c}_k\right)\Delta x},\left[-\pi ,\pi \right]$

Concept used:

  $\underset{n\to \infty }{\lim }{\displaystyle \sum _{k=1}^{n}f\left(c_k\right)\Delta x}={\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)dx}$

Calculation:

  $\begin{array}{l}\underset{n\to \infty }{\lim }{\displaystyle \sum _{k=1}^n\left({\sin }^3{c}_k\right)\Delta x},\left[-\pi ,\pi \right]\\ \Rightarrow f\left(c_k\right)={\sin }^3{c}_k\\ \Rightarrow f\left(x\right)={\sin }^{3}x\end{array}$

Therefore definite integral is,

  ${\displaystyle \underset{-\pi }{\overset{\pi }{\int }}f\left(x\right)dx}={\displaystyle \underset{-\pi }{\overset{\pi }{\int }}{\sin }^{3}xdx}$