Trigonometric Integrals

Author: John J Weber III, PhD Corresponding Textbook Sections:

• Section 8.1 – Arclength

Expected Educational Results

• Objective 12–1: I can set up the integral to compute the length of a curve.
• Objective 12–2: I can use technology to estimate the integral to compute the length of a curve.

Bloom’s Taxonomy

A modern version of Bloom’s Taxonomy is included here to recognize various different levels of understanding and to encourage you to work towards higher-order understanding (those at the top of the pyramid). All Objectives, Investigations, Activities, etc. are color-coded with the level of understanding.

Arclength

Bloom’s Taxonomy

A modern version of Bloom’s Taxonomy is included here to recognize various different levels of understanding and to encourage you to work towards higher-order understanding (those at the top of the pyramid). All Objectives, Investigations, Activities, etc. are color-coded with the level of understanding.

Definitions

• If $f^{\prime }(x)$$f^{\,\prime}(x)$ is continuous on $[a,b]$$[a,b]$, then the length of the curve $y=f(x)$$y=f(x)$ for $a\le x\le b$$a\leq x\leq b$ is $L={\int }_a^b\sqrt{1+(\frac{dy}{dx}{)}^2}dx$$L=\int_a^b{\sqrt{1+\Big(\frac{dy}{dx}\Big)^2}\,\,dx}$
• If $g^{\prime }(y)$$g^{\,\prime}(y)$ is continuous for $c\le y\le d$$c\leq y\leq d$, then the length of the curve $x=g(y)$$x=g(y)$ is $L={\int }_c^d\sqrt{1+(\frac{dx}{dy}{)}^2}dy$$L=\int_c^d{\sqrt{1+\Big(\frac{dx}{dy}\Big)^2}\,\,dy}$
• If $f^{\prime }(t)$$f^{\,\prime}(t)$ and $g^{\prime }(t)$$g^{\,\prime}(t)$ are continuous for $r\le t\le s$$r\leq t\leq s$, then the length of the parametric curve $x=f(t)$$x=f(t)$, $y=g(t)$$y=g(t)$ is $L={\int }_r^s\sqrt{(\frac{dx}{dt}{)}^2+(\frac{dy}{dt}{)}^2}dt$$L=\int_r^s{\sqrt{\Big(\frac{dx}{dt}\Big)^2+\Big(\frac{dy}{dt}\Big)^2}\,\,dt}$
• We will discuss Arclength of Polar Curves later in the course.

Investigation 01

Find the exact arclength:

1. $y=x^{\frac{3}{2}}$$y=x^{\frac{3}{2}}$ on $[1,4]$$[1,4]$.
2. $x=3-2y$$x=3-2y$ from $y=-1$$y=-1$ to $y=2$$y=2$.

Investigation 02

Set up the integral to find the exact arclength:

1. $x=y^2$$x=y^2$ from $y=0$$y=0$ to $y=4$$y=4$.
2. $y=\sin (x)$$y=\sin{(x)}$ from $x=0$$x=0$ to $x=\pi$$x=\pi$.
3. $y=\sqrt{x}$$y=\sqrt{x}$ on $[4,9]$$[4,9]$.
4. $y=3-x+x^2$$y=3-x+x^2$ from $x=1$$x=1$ to $x=2$$x=2$.
5. $x=\tan (y)$$x=\tan{(y)}$ from $y=\frac{\pi }{6}$$y=\frac{\pi}{6}$ to $y=\frac{\pi }{4}$$y=\frac{\pi}{4}$.
6. $x=\sqrt[3]{y}$$x=\sqrt[3]{y}$ on from $y=2$$y=2$ to $y=5$$y=5$.
7. $y=\sin (2x)$$y=\sin{(2x)}$ from $x=0$$x=0$ to $x=\pi$$x=\pi$.
8. $x=\ln (y)$$x=\ln{(y)}$ on from $y=2$$y=2$ to $y=5$$y=5$.
9. $y={\tan }^{-1}(x)$$y=\tan^{-1}{(x)}$ from $x=-2$$x=-2$ to $x=3$$x=3$.
10. $x=e^{5y}$$x=e^{5y}$ on from $y=-1$$y=-1$ to $y=1$$y=1$.

Investigation 03

1. Find the length of the curve defined by the parametric equations $x=t$$x=t$, $y=\sqrt{t}$$y=\sqrt{t}$ from $t=0$$t=0$ to $t=10$$t=10$.
2. Find the length of the curve defined by the parametric equations $x=\cos (2t)$$x=\cos{(2t)}$, $y=t^2+1$$y=t^2+1$ from $t=\pi$$t=\pi$ to $t=2\pi$$t=2\pi$.
3. Find the length of the curve defined by the parametric equations $x=e^{-3t}$$x=e^{-3t}$, $y=\ln (t)$$y=\ln{(t)}$ from $t=3$$t=3$ to $t=7$$t=7$.

Investigation 04

Use technology to estimate (to two (2) decimal places) any of the arclengths from Investigation 02 or Investigation 03.

Use Technology to Approximate Arclength

NOTE: On all Assessments, if you are asked for a decimal approximation for arclength, you may use technology to estimate the arclength without showing any work.

Mathematica

​x
1
(* Estimate the arclength of x=cos(3y) from y=0 to y=pi *)
2
NIntegrate[Sqrt[(-3Sin[3y])^2+1], {y, 0, Pi}]
3

4
(* Alternatively *)
5
Integrate[Sqrt[(-3Sin[3y])^2+1], {y, 0, Pi}]//N

Warnings:

1. Be very careful with the syntax. Syntax is the set of rules on how to write computer code. Every software program has its own unique syntax. Some basic Mathematica syntax is located at: https://www.jjw3.com/Common_Mathematica_Code.html.

2. NIntegrate[ ] has two arguments:

   1. The function in the argument of the integral;

   2. $\{x,a,b\}$$\{x,a,b\}$, where

      1. $x$$x$ is the independent variable;
      2. $a$$a$ is the lower limit of integration;
      3. $b$$b$ is the upper limit of integration.

3. For help on using the Integrate[ ] function:

   1. In Mathematica, execute the code: $\text{?NIntegrate}$$\text{?NIntegrate}$
   2. Click on $\vee$$\or$ near the bottom-left of output
   3. Click on local
   4. Read how to use the NIntegrate[ ] function – you will be able to copy-paste code.

4. Mathematica syntax:

   1. $\text{Sqrt[}\text{]}$$\text{Sqrt[\,\,]}$ represents $\sqrt{}$$\sqrt{\,\,}$
   2. $\text{E}\wedge \text{x}$$\text{E}\!\wedge\!\text{x}$ represents $e^x$$e^x$

5. When $\text{//N}$$\text{//N}$ is used at the end of the code, Mathematica will return a decimal approximation.

6. Remember, correct Mathematica code will be all black except for variables.

7. Parens, $($$($ and $)$$)$, may be needed to group terms.

8. To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.

Homework

At this time, you should be able to complete the following assignments:

• Section 8.1: # 1, 3, 5, 7, 9, 27, 41, 43.

CC BY-NC-SA 4.0

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License [http://creativecommons.org/licenses/by-nc-sa/4.0/].

Created: Tuesday, 29 September 2020 12:20 EDT Last Modified: Wednesday, 02 March 2022 - 03:19 (EST)