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Find the integral for the arc length of <math>x^2 + y^2 = 4</math> going clockwise from <math>(-3^{0.5}, 1)</math> to <math>(3^{0.5}, 1)</math>. | |||
The function is <math>y = \sqrt{4-x^2}</math>. <math>x</math> is ranging from <math>-\sqrt{3}</math> to <math>\sqrt3</math>. | |||
<cmath> | |||
\frac{dy}{dx} = -x \frac1{\sqrt{4-x^2}}. | |||
</cmath> | |||
Therefore the integral is | |||
<cmath> | |||
\boxed{\int_{-\sqrt3}^{\sqrt3} \sqrt{1+ \frac{x^2}{4-x^2}} \, dx}. | |||
</cmath> | |||
'''IGNORE EVERYTHING BELOW:''' | |||
<asy> | <asy> | ||
draw((0,0)--(0,2)--(2,2)--(2,0)--cycle); | draw((0,0)--(0,2)--(2,2)--(2,0)--cycle); | ||
Revision as of 16:41, 28 November 2008
Find the integral for the arc length of 
going clockwise from 
to 
.
The function is 
. 
is ranging from 
to 
.
![\[\frac{dy}{dx} = -x \frac1{\sqrt{4-x^2}}.\]](http://latex.artofproblemsolving.com/1/0/6/1069c675015e030962604ee5e6e667516dc6fd38.png)
Therefore the integral is
![\[\boxed{\int_{-\sqrt3}^{\sqrt3} \sqrt{1+ \frac{x^2}{4-x^2}} \, dx}.\]](http://latex.artofproblemsolving.com/7/c/9/7c9f2bb4e3555933b19ff567af5f71cfc510429e.png)
IGNORE EVERYTHING BELOW:
![[asy] draw((0,0)--(0,2)--(2,2)--(2,0)--cycle); draw((0,0)--(0,1)--(1,0)--cycle); draw((0,0)--(0,2)--(1,2)--cycle); draw((0,0)--(2,0)--(2,1)--cycle); fill((0,0)--(0,1)--(1,0)--cycle,gray); fill((0,0)--(0,2)--(1,2)--cycle,gray); fill((0,0)--(2,0)--(2,1)--cycle,gray); label("0",(0,0),SW); label("1",(0,1),W); label("2",(0,2),NW); label("1",(1,0),S); label("2",(2,0),SE); [/asy]](http://latex.artofproblemsolving.com/3/0/a/30a2f3d0fad996ac51c6e5c23b8e90a93b7e30f2.png)
![[asy] draw((0,0)--(5,12)); draw((5,12)--(8,12)); draw((0,0)--(13,0)); draw((8,12)--(13,0)); draw((5/3,4)--(13/3,0)); draw((13-5/3,4)--(13-13/3,0)); draw((10/3,8)--(6,12)); draw((7,12)--(13-10/3,8)); dot((5,12)); dot((8,12)); dot((13,0)); dot((0,0)); dot((5/3,4)); dot((13/3,0)); dot((13-5/3,4)); dot((13-13/3,0)); dot((10/3,8)); dot((13-10/3,8)); dot((6,12)); dot((7,12)); dot((6+1/3,12)); dot((7-1/3,12)); dot((5/3+5/9,4+4/3)); dot((5/3+10/9,4+8/3)); dot((13-5/3-5/9,4+4/3)); dot((13-5/3-10/9,4+8/3)); dot((13/3 + 26/9,0)); dot((13/3 + 13/9,0)); label("A",(5,12),NW); label("B",(8,12),NE); label("C",(13,0),SE); label("D",(0,0),SW); [/asy]](http://latex.artofproblemsolving.com/e/a/d/ead8a4738a16d570f97b9c23b183285f1e0a54c3.png)
![[asy] draw((0,0)--(5,12)); draw((5,12)--(8,12)); draw((0,0)--(13,0)); draw((5,12)--(5,0)); draw((8,12)--(13,0)); draw((8,12)--(8,0)); dot((5,12)); dot((8,12)); dot((13,0)); dot((0,0)); dot((5,0)); dot((8,0)); label("A",(5,12),NW); label("B",(8,12),NE); label("C",(13,0),SE); label("D",(0,0),SW); label("M",(5,0),S); label("N",(8,0),S); label("13",(2.5,6),NW); label("13",(11.5,6),NE); label("13",(6.5,-1),S); label("3",(6.5,12),N); label("12",(5,6),W); label("12",(8,6),E); label("5",(2.5,0),N); label("5",(10.5,0),N); [/asy]](http://latex.artofproblemsolving.com/c/3/2/c32ab0350f79fd4a3158a7614d1a894232ebd0dc.png)
