Find the arc length of the graph of the given equation on the specified interval.
$$y = \frac { 2 } { 3 } \left( x ^ { 2 } + 1 \right) ^ { 3 / 2 } , [ 2,3 ]$$
A) $\frac { 41 } { 3 }$
B) $\frac { 41 } { 2 }$
C) $\frac { 41 } { 4 }$
D) $\frac { 41 } { 6 }$

Question

Quizplus · Accepted Answer

Explanation:The arc length of the graph of the given equation on the specified interval is given by the integral of the square root of 1 plus the square of the derivative of the function with respect to x. So,$$s = \int_2^3 \sqrt{1 + \left( \frac{d}{dx} \left[ \frac{2}{3} (x^2 + 1)^{3/2} \right] \right)^2} dx$$$$= \int_2^3 \sqrt{1 + \left( \frac{2}{3} \cdot \frac{3}{2} (x^2 + 1)^{1/2} \cdot 2x \right)^2} dx$$$$= \int_2^3 \sqrt{1 + 4x^2 (x^2 + 1)} dx$$$$= \int_2^3 \sqrt{4x^4 + 4x^2 + 1} dx$$$$= \int_2^3 \sqrt{(2x^2 + 1)^2} dx$$$$= \int_2^3 (2x^2 + 1) dx$$$$= \left[ \frac{2x^3}{3} + x \right]_2^3$$$$= \frac{2(3)^3}{3} + 3 - \frac{2(2)^3}{3} - 2$$$$= \frac{41}{3}$$

Find the Arc Length of the Graph of the Given y=23(x2+1)3/2,[2,3]y = \frac { 2 } { 3 } \left( x ^ { 2 } + 1 \right) ^ { 3 / 2 } , [ 2,3 ]y=32​(x2+1)3/2,[2,3]

Find the Arc Length of the Graph of the Given $y = \frac { 2 } { 3 } \left( x ^ { 2 } + 1 \right) ^ { 3 / 2 } , [ 2,3 ]$