Question

Determine length of the curve $y=x^{\frac{3}{2}}$ from $x=0$ to $x=4$

• A. $\frac{80\sqrt{1}0}{27}$
• B. $\frac{80\sqrt{2}0}{27}$
• C. $\frac{8\sqrt{1}0}{27}$
• D. None of these

Answer 1

The correct option is D None of these

The given function will be $f\left(x\right)=x^{\frac{3}{2}}$

$f^{\prime }\left(x\right)=\frac{3}{2}{x}^{\frac{1}{2}}$

The length of the curve will be $={\int }_b^a\sqrt{(\frac{3\sqrt{x}}{2}{)}^2+1}$

$=\frac{1}{2}{\int }_0^4\sqrt{9x+4}$

$=\frac{1}{27}\left(9\right(4)+4{)}^{3/2}-\frac{1}{2}(9\left(0\right)+4{)}^{3/2}$

$=\frac{80\sqrt{10}}{27}-\frac{8}{27}$