Chapter 8.4, Problem 2E

a.

To determine

To Find: The length of the curve.

Answer to Problem 2E

The length of the curve $\approx 2.057$ .

Explanation of Solution

Given information: curve $y=\tan x,-\frac{\pi }{3}\le x\le 0$

Formula used: if a smooth curve begins at $\left(a,c\right)$ and ends at $\left(b,d\right)$

  $a<b,c>d,$ then the length of the curve is

  $L={\displaystyle \underset{a}{\overset{b}{\int }}\sqrt{1+{\left(\frac{dy}{dx}\right)}^2}}dx$

If y is a smooth function of x on $\left[a,b\right]$

  $L={\displaystyle \underset{c}{\overset{d}{\int }}\sqrt{1+{\left(\frac{dx}{dy}\right)}^2}}dy$

If x is a smooth function of y on $\left[c,d\right]$

Calculation:

curve $y=\tan x,-\frac{\pi }{3}\le x\le 0$

differentiate both sides,

  $\frac{dy}{dx}={\sec }^{2}x$

Which is continuous on $\left[-\frac{\pi }{3},0\right]$ .

Therefore

  $\begin{array}{l}L={\displaystyle \underset{\frac{-\pi }{3}}{\overset{0}{\int }}\sqrt{1+{\left({\sec }^{2}x\right)}^2}}dx\\ L\approx 2.057\end{array}$

b.

To determine

To find the graph of the curve of $y=\tan x$ .

Explanation of Solution

Given information: curve $y=\tan x,-\frac{\pi }{3}\le x\le 0$ .

Graph:

Use graphing utility to graph the curve,

  

c.

To determine

To Find: Use NINT to find the length of the curve.

Answer to Problem 2E

The length of the curve $\approx 2.057$ .

Explanation of Solution

Given information: curve $y=\tan x,-\frac{\pi }{3}\le x\le 0$

Formula used: if a smooth curve begins at $\left(a,c\right)$ and ends at $\left(b,d\right)$

  $a<b,c>d,$ then the length of the curve is

  $L={\displaystyle \underset{a}{\overset{b}{\int }}\sqrt{1+{\left(\frac{dy}{dx}\right)}^2}}dx$

If y is a smooth function of x on $\left[a,b\right]$

  $L={\displaystyle \underset{c}{\overset{d}{\int }}\sqrt{1+{\left(\frac{dx}{dy}\right)}^2}}dy$

If x is a smooth function of y on $\left[c,d\right]$

Calculation:

curve $y=\tan x,-\frac{\pi }{3}\le x\le 0$

  $\begin{array}{l}L={\displaystyle \underset{\frac{-\pi }{3}}{\overset{0}{\int }}\sqrt{1+{\left({\sec }^{2}x\right)}^2}}dx\\ L\approx 2.057\end{array}$

By NINT