$\text { Find the arc length of the graph of the
function } y = \frac { 2 } { 3 } x ^ { \frac { 3 } { 2 } } + 2 \text { over the interval [14, }$ $16$].
A) $17 \sqrt { 17 } - 15 \sqrt { 15 }$
B) $\frac { 2 } { 3 } ( 17 \sqrt { 17 } - 15 \sqrt { 15 } )$
C) $\frac { 3 } { 2 } ( 17 \sqrt { 17 } + 15 \sqrt { 15 } )$
D) $\frac { 3 } { 2 } ( 17 \sqrt { 17 } - 15 \sqrt { 15 } )$
E) $\frac { 2 } { 3 } ( 17 \sqrt { 17 } + 15 \sqrt { 15 } )$

Question

$\text { Find the arc length of the graph of the 
function } y = \frac { 2 } { 3 } x ^ { \frac { 3 } { 2 } } + 2 \text { over the interval [14, }$ $16$]. 
A) $17 \sqrt { 17 } - 15 \sqrt { 15 }$
B) $\frac { 2 } { 3 } ( 17 \sqrt { 17 } - 15 \sqrt { 15 } )$
C) $\frac { 3 } { 2 } ( 17 \sqrt { 17 } + 15 \sqrt { 15 } )$
D) $\frac { 3 } { 2 } ( 17 \sqrt { 17 } - 15 \sqrt { 15 } )$
E) $\frac { 2 } { 3 } ( 17 \sqrt { 17 } + 15 \sqrt { 15 } )$

Quizplus · Accepted Answer

The arc length $L$ of a function $y = f(x)$ over an interval $[a, b]$ is given by $L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} dx$. For $y = \frac{2}{3}x^{\frac{3}{2}} + 2$, $\frac{dy}{dx} = x^{\frac{1}{2}}$. Thus, $L = \int_{14}^{16} \sqrt{1 + x} dx$, which simplifies to $\frac{2}{3}(17\sqrt{17} - 15\sqrt{15})$, making option B correct.