use sigma notation to write the following riemann sum. then, evaluate the riemann sum using formulas for the…

use sigma notation to write the following riemann sum. then, evaluate the riemann sum using formulas for the sums of powers of positive integers or a calculator. the right riemann sum for f(x)=x + 1 on 0,4 with n = 30. a. ∑k = 130215k - 1 b. ∑k = 130215k + 1215 c. ∑k = 130215k - 1215 d. ∑k = 130215k+215 the right riemann sum is. (round to two decimal places as needed.)

use sigma notation to write the following riemann sum. then, evaluate the riemann sum using formulas for the sums of powers of positive integers or a calculator. the right riemann sum for f(x)=x + 1 on 0,4 with n = 30. a. ∑k = 130215k - 1 b. ∑k = 130215k + 1215 c. ∑k = 130215k - 1215 d. ∑k = 130215k+215 the right riemann sum is. (round to two decimal places as needed.)

Answer

Explanation:

Step1: Find $\Delta x$

For $a = 0$, $b = 4$ and $n=30$, $\Delta x=\frac{b - a}{n}=\frac{4-0}{30}=\frac{2}{15}$.

Step2: Find $x_k$

The right - hand endpoint of the $k$th sub - interval is $x_k=a + k\Delta x=0+\frac{2}{15}k=\frac{2}{15}k$.

Step3: Find $f(x_k)$

Given $f(x)=x + 1$, then $f(x_k)=\frac{2}{15}k+1$.

Step4: Write the right - Riemann sum in sigma notation

The right - Riemann sum $\sum_{k = 1}^{n}f(x_k)\Delta x$. Substituting $n = 30$, $f(x_k)=\frac{2}{15}k + 1$ and $\Delta x=\frac{2}{15}$, we get $\sum_{k=1}^{30}\left[\frac{2}{15}k + 1\right]\frac{2}{15}$, so the correct option for the sigma notation is B.

Step5: Evaluate the Riemann sum

[ \begin{align*} \sum_{k = 1}^{30}\left[\frac{2}{15}k+1\right]\frac{2}{15}&=\frac{2}{15}\sum_{k = 1}^{30}\left(\frac{2}{15}k + 1\right)\ &=\frac{2}{15}\left(\frac{2}{15}\sum_{k=1}^{30}k+\sum_{k = 1}^{30}1\right) \end{align*} ] We know that $\sum_{k=1}^{n}k=\frac{n(n + 1)}{2}$ and $\sum_{k=1}^{n}1=n$. For $n = 30$, $\sum_{k=1}^{30}k=\frac{30\times(30 + 1)}{2}=465$ and $\sum_{k=1}^{30}1=30$. [ \begin{align*} \frac{2}{15}\left(\frac{2}{15}\times465+30\right)&=\frac{2}{15}\left(\frac{930}{15}+30\right)\ &=\frac{2}{15}(62 + 30)\ &=\frac{2}{15}\times92\ &=\frac{184}{15}\approx12.27 \end{align*} ]

Answer:

B. $\sum_{k = 1}^{30}\left[\frac{2}{15}k + 1\right]\frac{2}{15}$ 12.27