Arc Length - Revision history

Lila: /* Resources */

2021-11-02T16:15:10Z

Resources

2021-10-01T20:01:55Z

Resources

2021-10-01T20:00:50Z

Example Problem

2021-10-01T20:00:12Z

2021-10-01T19:58:01Z

2021-09-26T02:18:45Z

Created page with "* [https://openstax.org/books/calculus-volume-3/pages/3-3-arc-length-and-curvature Arc Length and Curvature], OpenStax"

New page

* [https://openstax.org/books/calculus-volume-3/pages/3-3-arc-length-and-curvature Arc Length and Curvature], OpenStax

← Older revision		Revision as of 16:15, 2 November 2021
Line 30:		Line 30:
	* [https://en.wikibooks.org/wiki/Calculus/Arc_length Arc Length], WikiBooks: Calculus		* [https://en.wikibooks.org/wiki/Calculus/Arc_length Arc Length], WikiBooks: Calculus
	* [https://openstax.org/books/calculus-volume-3/pages/3-3-arc-length-and-curvature Arc Length and Curvature], OpenStax		* [https://openstax.org/books/calculus-volume-3/pages/3-3-arc-length-and-curvature Arc Length and Curvature], OpenStax
		+
		+	==Licensing==
		+	Content obtained and/or adapted from:
		+	* [https://en.wikibooks.org/wiki/Calculus/Arc_length Arc Length, WikiBooks: Calculus] under a CC BY-SA license

← Older revision		Revision as of 20:01, 1 October 2021
Line 28:		Line 28:

	==Resources==		==Resources==
		+	* [https://en.wikibooks.org/wiki/Calculus/Arc_length Arc Length], WikiBooks: Calculus
	* [https://openstax.org/books/calculus-volume-3/pages/3-3-arc-length-and-curvature Arc Length and Curvature], OpenStax		* [https://openstax.org/books/calculus-volume-3/pages/3-3-arc-length-and-curvature Arc Length and Curvature], OpenStax

@@ Line 6: / Line 6: @@
 For those who prefer simplicity, the formula can be rewritten into:<blockquote><math>L=\int_a^b|\mathbf{r}'(t)|dt\quad </math> or <math>\quad\frac{dL}{dt}=|\mathbf{r}'(t)|</math></blockquote>
-===Example Problem===
+===Example Problems===
-Find the circumference of the circle given by the parametric equations <math>x(t)=R\cos(t),y(t)=R\sin(t)</math> , with <math>t\in[0,2\pi]</math>.
+. Find the circumference of the circle given by the parametric equations <math>x(t)=R\cos(t),y(t)=R\sin(t)</math> , with <math>t\in[0,2\pi]</math>.
 :<math>\begin{align}s&=\int\limits_0^{2\pi}\sqrt{\left(\tfrac{d}{dt}\big(R\cos(t)\big)\right)^2+\left(\tfrac{d}{dt}\big(R\sin(t)\big)\right)^2}dt\\
@@ Line 15: / Line 15: @@
 &=R\cdot t\Big|_0^{2\pi}\\
 &=\mathbf{2\pi R}\end{align}</math>
 ==Resources==
 * [https://openstax.org/books/calculus-volume-3/pages/3-3-arc-length-and-curvature Arc Length and Curvature], OpenStax

@@ Line 1: / Line 1: @@
 We can deduce that the length of a curve with parametric equations <math>\begin{cases} x=f(t) \\ y=g(t) \end{cases} </math>, <math>a\le t\le b </math> should be:<blockquote><math>L=\int_a^b\sqrt{\biggl(\frac{dx}{dt}\biggr)^2+\biggl(\frac{dy}{dt}\biggr)^2}dt </math></blockquote>Since vector functions are fundamentally parametric equations with directions, we can utilize the formula above into the length of a space curve.
-===Arc length of a space curve===
+==Arc length of a space curve==
 If the curve has the vector equation <math>\mathbf{r}(t)=\langle f(t),g(t),h(t)\rangle,a\le t\le b</math>, or, equivalently, the parametric equations <math>x=f(t),y=g(t),z=h(t)</math>, where <math>f',g',h'</math> are continuous, then the length of the curve from <math>t=a</math> to <math>t=b</math> is:
 :<math>L=\int_a^b\sqrt{[f'(t)]^2+[g'(t)]^2+[h'(t)]^2}dt=\int_a^b\sqrt{\biggl(\frac{dx}{dt}\biggr)^2+\biggl(\frac{dy}{dt}\biggr)^2+\biggl(\frac{dx}{dz}\biggr)^2}dt</math>}}
 For those who prefer simplicity, the formula can be rewritten into:<blockquote><math>L=\int_a^b|\mathbf{r}'(t)|dt\quad </math> or <math>\quad\frac{dL}{dt}=|\mathbf{r}'(t)|</math></blockquote>
 ==Resources==
 * [https://openstax.org/books/calculus-volume-3/pages/3-3-arc-length-and-curvature Arc Length and Curvature], OpenStax

← Older revision		Revision as of 19:58, 1 October 2021
Line 1:		Line 1:
		+	We can deduce that the length of a curve with parametric equations <math>\begin{cases} x=f(t) \\ y=g(t) \end{cases} </math>, <math>a\le t\le b </math> should be:<blockquote><math>L=\int_a^b\sqrt{\biggl(\frac{dx}{dt}\biggr)^2+\biggl(\frac{dy}{dt}\biggr)^2}dt </math></blockquote>Since vector functions are fundamentally parametric equations with directions, we can utilize the formula above into the length of a space curve.
		+	===Arc length of a space curve===
		+
		+	If the curve has the vector equation <math>\mathbf{r}(t)=\langle f(t),g(t),h(t)\rangle,a\le t\le b</math>, or, equivalently, the parametric equations <math>x=f(t),y=g(t),z=h(t)</math>, where <math>f',g',h'</math> are continuous, then the length of the curve from <math>t=a</math> to <math>t=b</math> is:
		+	:<math>L=\int_a^b\sqrt{[f'(t)]^2+[g'(t)]^2+[h'(t)]^2}dt=\int_a^b\sqrt{\biggl(\frac{dx}{dt}\biggr)^2+\biggl(\frac{dy}{dt}\biggr)^2+\biggl(\frac{dx}{dz}\biggr)^2}dt</math>}}
		+	For those who prefer simplicity, the formula can be rewritten into:<blockquote><math>L=\int_a^b\|\mathbf{r}'(t)\|dt\quad </math> or <math>\quad\frac{dL}{dt}=\|\mathbf{r}'(t)\|</math></blockquote>
		+
		+	==Resources==
	* [https://openstax.org/books/calculus-volume-3/pages/3-3-arc-length-and-curvature Arc Length and Curvature], OpenStax		* [https://openstax.org/books/calculus-volume-3/pages/3-3-arc-length-and-curvature Arc Length and Curvature], OpenStax