Write the limit $\lim _ { | \Delta x | \rightarrow 0 } \sum _ { i = 1 } ^ { n } 4 c _ { i } \left( 7 - c _ { i } \right) ^ { 2 } \Delta x _ { i } , \text { as a definite integral on the interval }$ $[ 0,8 ]$ where $c _ { i }$ is any point in the $i ^ { t h }$ subinterval.
A) $ \int 4 x(7-x)^{8} d x $
B) $ \int_{8}^{0} 4 x^{2}(7-x)^{8} d x $
C) $ \int 4 c_{i}\left(7-c_{i}\right)^{2} \Delta x_{i} $
D)$ \int_{0}^{8} 4 x(7-x)^{2} d x $
E)$ \int_{0}^{c_{i}} 4 c_{i}\left(7-c_{i}\right)^{2} \Delta x_{i} $

Question

Quizplus · Accepted Answer

The given limit can be written as:
$$\lim _ { | \Delta x | \rightarrow 0 } \sum _ { i = 1 } ^ { n } 4 c _ { i } \left( 7 - c _ { i } \right) ^ { 2 } \Delta x _ { i } = \int_{0}^{8} 4 x(7-x)^{2} d x$$
This is because, as $|\Delta x| \rightarrow 0$, the Riemann sum approaches the definite integral. Also, the expression $4c_i(7-c_i)^2$ is continuous on the interval $[0,8]$, so we can evaluate it at any point in the subinterval. Thus, the definite integral on the interval $[0,8]$ that represents the given limit is $\int_{0}^{8} 4 x(7-x)^{2} d x$.

Write the Limit [0,8][ 0,8 ][0,8] Where cic _ { i }ci​

Write the Limit $[ 0,8 ]$ Where $c _ { i }$