Given $f ( x ) = \ln ( x + 1 ) ; \quad 0 \leq x \leq 1 , n = 3 ,$ set up a Riemann sum to approximate the area under the graph of f(x) on the given interval. Use the midpoints. Is the following the correct answer?
$\ln \left( \frac { 7 } { 6 } \right) + \ln \left( \frac { 3 } { 2 } \right) + \ln \left( \frac { 11 } { 6 } \right)$

Question

Quizplus · Accepted Answer

To set up a Riemann sum using midpoints for $n=3$ on the interval $[0,1]$, the interval is divided into 3 equal parts, each of width $\Delta x = \frac{1-0}{3} = \frac{1}{3}$. The midpoints of these intervals are $x_1 = \frac{1}{6}, x_2 = \frac{1}{2},$ and $x_3 = \frac{5}{6}$. The correct Riemann sum is $f\left(\frac{1}{6}\right)\Delta x + f\left(\frac{1}{2}\right)\Delta x + f\left(\frac{5}{6}\right)\Delta x = \ln\left(\frac{7}{6}\right)\frac{1}{3} + \ln\left(\frac{3}{2}\right)\frac{1}{3} + \ln\left(\frac{11}{6}\right)\frac{1}{3}$, not the sum without the $\Delta x$ factors.

Given f(x)=ln⁡(x+1);0≤x≤1,n=3,f ( x ) = \ln ( x + 1 ) ; \quad 0 \leq x \leq 1 , n = 3 ,f(x)=ln(x+1);0≤x≤1,n=3,

Given $f ( x ) = \ln ( x + 1 ) ; \quad 0 \leq x \leq 1 , n = 3 ,$