Question

question 7 - 1 point which of the following answer choices represents the limit definition of the definite integral below using a right - endpoint riemann sum? (int_{-7}^{10}10x^{3}dx) select the correct answer below: (lim_{n
ightarrowinfty}\frac{170}{n^{4}}sum_{i = 1}^{n}(17i-7n)^{3}) (lim_{n
ightarrowinfty}\frac{340}{n^{4}}sum_{i = 1}^{n}(17i-7n)^{3}) (lim_{n
ightarrowinfty}\frac{170}{n^{4}}sum_{i = 1}^{n}(17i-7n - 17)^{3}) (lim_{n
ightarrowinfty}\frac{510}{n^{4}}sum_{i = 1}^{n}(17i-7n - 17)^{3})

Explanation:

Step1: Recall the limit - definition of definite integral

The definite integral $\int_{a}^{b}f(x)dx=\lim_{n
ightarrow\infty}\sum_{i = 1}^{n}f(x_i)\Delta x$, where $\Delta x=\frac{b - a}{n}$ and $x_i=a + i\Delta x$. Here, $a=-7$, $b = 10$, so $\Delta x=\frac{10-(-7)}{n}=\frac{17}{n}$ and $x_i=-7 + i\frac{17}{n}$. And $f(x)=10x^{3}$. Then $f(x_i)=10(-7 + i\frac{17}{n})^{3}$.

Step2: Calculate the Riemann - sum

$\sum_{i = 1}^{n}f(x_i)\Delta x=\sum_{i = 1}^{n}10(-7 + i\frac{17}{n})^{3}\frac{17}{n}$.
\[

$$\begin{align*} \lim_{n ightarrow\infty}\sum_{i = 1}^{n}10(-7 + i\frac{17}{n})^{3}\frac{17}{n}&=\lim_{n ightarrow\infty}\frac{170}{n^{4}}\sum_{i = 1}^{n}(17i-7n)^{3} \end{align*}$$

\]

Answer:

$\lim_{n
ightarrow\infty}\frac{170}{n^{4}}\sum_{i = 1}^{n}(17i - 7n)^{3}$