Q. 51 - Calculus

Question

Each definite integral represents the arc length of a function f(x) on an interval [a, b]. Determine the function and interval.

${\int }_0^{\mathrm{\pi }/4}secxdx$ 

Step-by-Step Solution

Answer

The function is $f\left(x\right)=\ln \left|\cos x\right|$ and interval is $\left[a,b\right]=\left[0,\frac{\mathrm{\pi }}{4}\right]$.

Step 1. Given information.

Consider the given function is ${\int }_0^{\mathrm{\pi }/4}secxdx$.

Step 2. Use arc length formula.

The formula for a function to find the arc length from $x=a$ to $x=b$ is given by  ${\int }_a^b\sqrt{1+{\left(f^{\text{'}}\left(x\right)\right)}^2}dx$.

Step 3. Find derivative of given function.

Compare given definite integral function with the arc length formula.

 $\begin{array}{rcl}{\int }_a^b\sqrt{1+{\left(f^{\text{'}}\left(x\right)\right)}^2}dx& =& {\int }_0^{\mathrm{\pi }/4}secxdx\\ & =& {\int }_0^{\mathrm{\pi }/4}\sqrt{1+{\left(\tan x\right)}^2}dx\end{array}$

Therefore the function  will be $f^{\text{'}}\left(x\right)=\begin{array}{rcl}& & \tan \end{array}\begin{array}{rcl}& & x\end{array}$ and interval is  $\left[a,b\right]=\left[0,\frac{\mathrm{\pi }}{4}\right]$.

Step 4. Find function f x .

The function whose differentiation is $\tan x$ is $\ln \left|\cos x\right|$.

Thus the required function is $f\left(x\right)=\ln \left|\cos x\right|$ and $\left[a,b\right]=\left[0,\frac{\mathrm{\pi }}{4}\right]$.