Answer
$\approx28.794$
Work Step by Step
The formula for the arc length is
$s=\int_{a}^b \sqrt{1 + (y')^2} dy$
Differentiate $y = \frac{3}{2}x^\frac{3}{2} + 4$ with respect to x.
$y' = \frac{1}{\sqrt[3]{x}}$
Substitute the value of y' in $1 + (y')^2$
$1+(y')^2=1+(\frac{1}{\sqrt[3]{x}})^2$
Substitute the value of $1 + (y')^2$ in the formula for the arc length $s=\int_{a}^b \sqrt{1 + (y')^2} dy$
Solve for the distance s
$s=\int_{1}^{27}\sqrt{1+(\frac{1}{\sqrt[3]{x}})^2} dx$
$=\int_{1}^{27}(\frac{1}{x^\frac{2}{3}}+1)^\frac{3}{2}xdx$
$\approx28.794$
