Q. 33 - 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 .

$f\left(x\right)=2x^{\frac{3}{2}}+1$ , $\left[1,3\right]$

Step-by-Step Solution

Answer

The arc length is 19 .

Step 1. Given information .

Consider the given function $f\left(x\right)=2x^{\frac{3}{2}}+1$ .

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 .

$$\begin{gathered} f\left(x\right)=2x^{\frac{3}{2}}+1 \\ f\text{'}\left(x\right)=\frac{d}{dx}\left[f\left(x\right)\right]=3x^{\frac{1}{2}} \\ Arc length={\int }_1^3\sqrt{1+{\left(3x^{\frac{1}{2}}\right)}^2}dx \\ ={\int }_1^3\sqrt{1+9x} dx \\ =\frac{1}{2}{\int }_1^3{\left(1+9x\right)}^{\frac{1}{2}} dx \\ =\frac{1}{2}{\int }_1^3\left(1+9x\right) dx \\ =\frac{1}{2}{\left[x+\frac{9x^2}{2}\right]}_1^3 \\ =\frac{1}{2}·\frac{76}{2} \\ =19 \end{gathered}$$