2. if $f(x)=x^{2}-2x$, $0leqslant xleqslant3$, evaluate the riemann sum with $n = 6$, taking the sample…

Explanation:

Step1: Calculate the width of sub - intervals

The interval is $[a,b]=[0,3]$ and $n = 6$. The width of each sub - interval $\Delta x=\frac{b - a}{n}=\frac{3-0}{6}=\frac{1}{2}$.

Step2: Determine the right - endpoints

The right - endpoints $x_i=a + i\Delta x$, for $i = 1,2,\cdots,6$. So $x_1=0.5,x_2 = 1,x_3=1.5,x_4 = 2,x_5=2.5,x_6=3$.

Step3: Evaluate the function at the right - endpoints

$f(x_1)=(0.5)^2-2\times0.5=0.25 - 1=- 0.75$; $f(x_2)=1^2-2\times1=1 - 2=-1$; $f(x_3)=(1.5)^2-2\times1.5=2.25-3=-0.75$; $f(x_4)=2^2-2\times2=4 - 4 = 0$; $f(x_5)=(2.5)^2-2\times2.5=6.25 - 5 = 1.25$; $f(x_6)=3^2-2\times3=9 - 6 = 3$.

Step4: Calculate the Riemann sum

The Riemann sum $R_6=\sum_{i = 1}^{6}f(x_i)\Delta x$. $R_6=\Delta x\left[f(x_1)+f(x_2)+f(x_3)+f(x_4)+f(x_5)+f(x_6)\right]$ $=\frac{1}{2}(-0.75-1 - 0.75+0+1.25+3)$ $=\frac{1}{2}(1.75)=\frac{7}{8}=0.875$. The Riemann sum with right - endpoints represents the sum of the areas of rectangles, where the height of each rectangle is given by the value of the function $y = f(x)$ at the right - hand endpoint of each sub - interval and the width of each rectangle is $\Delta x$.

Answer:

The value of the Riemann sum is $0.875$. It represents the sum of the areas of rectangles with heights determined by the function values at the right - endpoints of the sub - intervals and width $\Delta x=\frac{1}{2}$ over the interval $[0,3]$.