Q. 56 - Calculus

Question

You may have noticed that even very simple functions give rise to arc length integrals that we have no idea how to compute. In Exercises 53–56, use a graphing calculator to approximate a definite integral that represents the arc length of the given function f(x) on the interval [a, b].

$f\left(x\right)=\ln x,[a,b]=[1,3]$

Step-by-Step Solution

Answer

By using a graphing calculator the approximate arc length of the given function on the given interval is $2.30199.$

Step 1. Given Information.

The given function is $f\left(x\right)=\ln x$ and the interval is $\left[1,3\right].$

Step 2. Find the arc length.

We have to use a graphing calculator to approximate a definite integral that represents the arc length of the given function $f\left(x\right)=\ln x$ on the interval $\left[1,3\right].$

So, the arc length of a function from the given interval is given by ${\int }_a^b\sqrt{1+{(f\text{'}(x\left)\right)}^2}dx.$ 

Thus, the arc length of the given function on the given interval is $$\begin{gathered} {\int }_1^3\sqrt{1+{\left(\frac{1}{x}\right)}^2}dx \\ ={\int }_1^3\sqrt{1+\frac{1}{x^2}}dx \end{gathered}$$

Now, by using the graphing calculator the approximate arc length of the definite integral is ${\int }_1^3\sqrt{1+\frac{1}{x^2}}dx\approx 2.30199.$