Q. 34 - Calculus

Question

Find the exact value of the arc length of each function f (x)  on [a, b] by writing the arc length as a definite integral and then solving that integral .

$f\left(x\right)=x^{\frac{3}{2}}$ , $\left[a,b\right]=\left[0,2\right]$

Step-by-Step Solution

Answer

The arc length is $\frac{13}{4}$ .

Step 1. Given information .

Consider the given function $f\left(x\right)=x^{\frac{3}{2}}$ .

Step 2. Formula used .

The formula used to find the arc length from $x=a$ to $x=b$ is ,

Arc length of $f\left(x\right)={\int }_a^b\sqrt{1+{\left(f\text{'}\left(x\right)\right)}^2}dx$ .

Step 3. Find the arc length .

$$\begin{gathered} f\left(x\right)=x^{\frac{3}{2}} \\ f\text{'}\left(x\right)=\frac{d}{dx}\left[f\left(x\right)\right]=\frac{3}{2}{x}^{\frac{1}{2}} \end{gathered}$$

$$\begin{gathered} Arc length ={\int }_0^2\sqrt{1+{\left(\frac{3}{2}{x}^{\frac{1}{2}}\right)}^2}dx \\ ={\int }_0^2\sqrt{1+\frac{9}{4}x} dx \\ ={\int }_0^2{\left(1+\frac{9}{4}x\right)}^{\frac{1}{2}} dx \\ =\frac{1}{2}{\int }_0^2\left(1+\frac{9}{4}x\right) dx \\ =\frac{1}{2}{\left[x+\frac{9x^2}{8}\right]}_0^2 \\ =\frac{1}{2}\left[2+\frac{9·4}{8}\right] \\ =\frac{13}{4} \end{gathered}$$