Write the following limit as a definite integral on the interval $[ 2,8 ] \text { where } c _ { i } \text { is any }$ point in the $i ^ { \text {th } } \text { subinterval. }$ $$\lim _ { | \Delta x | \rightarrow 0 } \sum _ { i = 1 } ^ { n } \sqrt { 3 c _ { i } ^ { 2 } + 4 c _ { i } } \Delta x _ { i }$$
A) $\int _ { 2 } ^ { 8 } \sqrt { 3 x ^ { 2 } + 4 x } d x$
B) $\int _ { 2 } ^ { 8 } \sqrt { 9 x ^ { 2 } + 16 x } d x$
C) $\int _ { 8 } ^ { 2 } \sqrt { 3 x ^ { 2 } + 4 x } d x$
D) $\int _ { 2 } ^ { 8 } \sqrt { 3 x + 4 } d x$
E) $\int _ { 2 } ^ { 8 } \sqrt { 6 x + 4 } d x$

Question

Write the following limit as a definite integral on the interval $[ 2,8 ] \text { where } c _ { i } \text { is any }$ point in the $i ^ { \text {th } } \text { subinterval. }$ $$\lim _ { | \Delta x | \rightarrow 0 } \sum _ { i = 1 } ^ { n } \sqrt { 3 c _ { i } ^ { 2 } + 4 c _ { i } } \Delta x _ { i }$$
A) $\int _ { 2 } ^ { 8 } \sqrt { 3 x ^ { 2 } + 4 x } d x$
B) $\int _ { 2 } ^ { 8 } \sqrt { 9 x ^ { 2 } + 16 x } d x$
C) $\int _ { 8 } ^ { 2 } \sqrt { 3 x ^ { 2 } + 4 x } d x$ 
D) $\int _ { 2 } ^ { 8 } \sqrt { 3 x + 4 } d x$
E) $\int _ { 2 } ^ { 8 } \sqrt { 6 x + 4 } d x$

Quizplus · Accepted Answer

This limit can be recognized as a Riemann sum for the function $\sqrt{3x^2+4x}$ on the interval $[2,8]$. Therefore, taking the limit as $\Delta x$ approaches 0 gives the definite integral $\int_2^8 \sqrt{3x^2+4x}dx$. Therefore, the answer is (A).

Write the Following Limit as a Definite Integral on the Interval