Question

Find the arc length of the curve y=2x^3/2 from x=0 to x=3

Answer 1

The arc length of a function is given by the equation:

`L\text{ = }\int ^b_a\sqrt[]{1+\mleft(f\prime(x)\mright)^2}dx`

f'(x) = 3x^(1/2)

Now, applying the arc length formula:

`\begin{gathered} L\text{ = }\int ^3_0\sqrt[]{1+(3x^{(1)/(2)})^2}dx \\ L\text{ = }\int ^3_0\sqrt[]{1+9x}dx;\text{ }substituting\text{ t = 1 + 9x} \\ L\text{ = }(1)/(9)\int ^3_0\sqrt[]{t}\text{ dt} \\ L\text{ = }(1)/(9)*\frac{2\sqrt[]{t}\text{ }*\textt}{3};\text{ }replacing\colon \\ L\text{ = }(1)/(9)*\frac{2\sqrt[]{1+9x}\text{ }*\text1+9x}{3};\text{ from 0 to 3; evaluating the limits:} \\ L\text{ = }\frac{112\sqrt[]{7}-2}{27}=\text{ }10.90 \end{gathered}`