Chapter 7.4, Problem 21E

To determine

To calculate: To determine the curve's length as indicated by the equation $y={\displaystyle \underset{0}{\overset{x}{\int }}\sqrt{\cos \left(2t\right)dt.}}$

Answer to Problem 21E

To determine how long the Calculus Fundamental Theorem and the Table of Integrals should be $1$

Explanation of Solution

Given Information: Calculate the give curve's length using the given formula.

  ${\displaystyle \underset{0}{\overset{x/4}{\int }}{\sqrt{1+\left(\frac{dy}{dx}\right)}}^{2}dx}$

Calculation:

To determine $y$ first derivative in terms of $x$ .

It is common knowledge to determine the first derivative of the given expression $y=y(x)$ as

  $\begin{array}{c}\frac{dy}{dx}=\frac{\left({\displaystyle \underset{0}{\overset{x}{\int }}\sqrt{\cos \left(2t\right)}dt}\right)}{dx}\\ =\sqrt{\cos \left(2x\right)}\end{array}$

Then the curve's length is determined by

  $length={\displaystyle \underset{0}{\overset{x/4}{\int }}\sqrt{1+\cos \left(2x\right)}}dx$

Assessing the resulting integral.

The trigonometric identity is used ${\cos }^2\left(x\right)=\frac{1+\cos \left(2x\right)}{2}$ the sub integral function is transformed as

  $\begin{array}{c}\sqrt{1+\cos \left(2x\right)}=\sqrt{2{\cos }^2\left(x\right)}\\ =\sqrt{2}\cos \left(x\right)\end{array}$

Where the last equality results from the positive values of $\cos \left(x\right)$ over the range $\left[0,\frac{\pi }{4}\right]$ .

Returning the outcome of the previous step's results

  $length={\displaystyle \underset{0}{\overset{\pi /4}{\int }}\sqrt{2}\cos \left(x\right)dx}$

As well as the Fundamental Theorem of Calculus, are all that are required to calculate the length

  $\begin{array}{c}length={\displaystyle \underset{0}{\overset{\pi /4}{\int }}\sqrt{2}\cos \left(x\right)dx}\\ =\sqrt{2\sin \left(x\right)}\left|{}_0^{\pi /4}\right.\\ =\sqrt{2.\frac{\sqrt{2}}{2}}\\ =1\end{array}$

To determine how long the Calculus Fundamental Theorem and the Table of Integrals should be $1$