Question

use the figures to calculate the left and right riemann sums for f on the given interval and the given value of n. f(x)=x + 5 on 2,7; n = 5 the left riemann sum is . (simplify your answer.) the right riemann sum is . (simplify your answer.)

Explanation:

Step1: Calculate the width of sub - intervals

The interval is $[a,b]=[2,7]$ and $n = 5$. The width of each sub - interval $\Delta x=\frac{b - a}{n}=\frac{7 - 2}{5}=1$.

Step2: Find the left - hand endpoints

The left - hand endpoints of the sub - intervals are $x_0 = 2,x_1=3,x_2 = 4,x_3=5,x_4 = 6$.

Step3: Calculate the left Riemann sum

The left Riemann sum $L_5=\sum_{i = 0}^{4}f(x_i)\Delta x$.
$f(x_i)=x_i + 5$.
$f(2)=2 + 5=7$, $f(3)=3 + 5=8$, $f(4)=4 + 5=9$, $f(5)=5 + 5=10$, $f(6)=6 + 5=11$.
$L_5=(7 + 8+9 + 10+11)\times1=45$.

Step4: Find the right - hand endpoints

The right - hand endpoints of the sub - intervals are $x_1 = 3,x_2=4,x_3 = 5,x_4=6,x_5 = 7$.

Step5: Calculate the right Riemann sum

The right Riemann sum $R_5=\sum_{i = 1}^{5}f(x_i)\Delta x$.
$f(3)=3 + 5=8$, $f(4)=4 + 5=9$, $f(5)=5 + 5=10$, $f(6)=6 + 5=11$, $f(7)=7 + 5=12$.
$R_5=(8 + 9+10 + 11+12)\times1=50$.

Answer:

The left Riemann sum is $45$.
The right Riemann sum is $50$.