Given the partition $\{0,\,\frac{\pi}{4},\,\frac{\pi}{2},\,\frac{3\pi}{4},\,\pi\}$, and points $\bar{x}_1=\frac{\pi}{8}$, $\bar{x}_2=\frac{3\pi}{8}$, $\bar{x}_3=\frac{5\pi}{8}$, and $\bar{x}_4=\frac{7\pi}{8}$; find $R(P)$ for $\int_0^{\pi}\sin(x)\,dx$.

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Lesson Parent: 
Integration IV: Definite Integrals As Riemann Sums
Problem: 

Given the partition $\{0,\,\frac{\pi}{4},\,\frac{\pi}{2},\,\frac{3\pi}{4},\,\pi\}$, and points $\bar{x}_1=\frac{\pi}{8}$, $\bar{x}_2=\frac{3\pi}{8}$, $\bar{x}_3=\frac{5\pi}{8}$, and $\bar{x}_4=\frac{7\pi}{8}$; find the Riemann sum $R(P)$ for the integral $\int_0^{\pi}\sin(x)\,dx$.

Answer: 

2.05

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Solution: 

This partition breaks the closed interval <Sign in to see all the formulas> into the intervals <Sign in to see all the formulas> where

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Each interval has a length denoted by <Sign in to see all the formulas> and given by:

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Next, let's determine the value of the sine function at each point <Sign in to see all the formulas>:

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We can now find that

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