Question

write the limit as a definite integral on the interval a, b, where ( c_i ) is any point in the ( i )th subinterval.
limit: ( lim_{||delta|| \to 0} sum_{i = 1}^{n} (7c_i + 13) delta x_i )
interval: ( -7, 3 )
______ ( dx )

Explanation:

Step1: Recall definite integral definition

The limit of a Riemann sum $\lim_{||\Delta|| \to 0} \sum_{i=1}^{n} f(c_i) \Delta x_i$ on $[a,b]$ equals $\int_{a}^{b} f(x) dx$.

Step2: Identify $f(x)$, $a$, and $b$

Here, $f(c_i)=7c_i + 13$, so $f(x)=7x+13$. The interval is $[-7,3]$, so $a=-7$, $b=3$.

Answer:

$\int_{-7}^{3} (7x + 13) dx$