User:Annoyingpi: Difference between revisions

Latest revision as of 00:30, 10 December 2008

$[asy] dot((-1,0)); dot((1,0)); dot((0,1.5)); dot((0,-1.5)); dot((0.3,0.3)); draw((1,0)--(0,1.5)--(-1,0)--(0,-1.5)--cycle); label("$P_1$",(-1,0),W); label("$P_2$",(1,0),E); label("$Q_1$",(0,1.5),N); label("$Q_2$",(0,-1.5),S); label("$X$",(0.3,0.3),SW); [/asy]$

$[asy] draw((0,0)--(0,2)--(2,2)--(2,0)--cycle); draw((0,0)--(0,1)--(1,0)--cycle); draw((0,0)--(0,2)--(1,2)--cycle); draw((0,0)--(2,0)--(2,1)--cycle); fill((0,0)--(0,1)--(1,0)--cycle,gray); fill((0,0)--(0,2)--(1,2)--cycle,gray); fill((0,0)--(2,0)--(2,1)--cycle,gray); label("0",(0,0),SW); label("1",(0,1),W); label("2",(0,2),NW); label("1",(1,0),S); label("2",(2,0),SE); [/asy]$

$[asy] draw((0,0)--(5,12)); draw((5,12)--(8,12)); draw((0,0)--(13,0)); draw((8,12)--(13,0)); draw((5/3,4)--(13/3,0)); draw((13-5/3,4)--(13-13/3,0)); draw((10/3,8)--(6,12)); draw((7,12)--(13-10/3,8)); dot((5,12)); dot((8,12)); dot((13,0)); dot((0,0)); dot((5/3,4)); dot((13/3,0)); dot((13-5/3,4)); dot((13-13/3,0)); dot((10/3,8)); dot((13-10/3,8)); dot((6,12)); dot((7,12)); dot((6+1/3,12)); dot((7-1/3,12)); dot((5/3+5/9,4+4/3)); dot((5/3+10/9,4+8/3)); dot((13-5/3-5/9,4+4/3)); dot((13-5/3-10/9,4+8/3)); dot((13/3 + 26/9,0)); dot((13/3 + 13/9,0)); label("A",(5,12),NW); label("B",(8,12),NE); label("C",(13,0),SE); label("D",(0,0),SW); [/asy]$

$[asy] draw((0,0)--(5,12)); draw((5,12)--(8,12)); draw((0,0)--(13,0)); draw((5,12)--(5,0)); draw((8,12)--(13,0)); draw((8,12)--(8,0)); dot((5,12)); dot((8,12)); dot((13,0)); dot((0,0)); dot((5,0)); dot((8,0)); label("A",(5,12),NW); label("B",(8,12),NE); label("C",(13,0),SE); label("D",(0,0),SW); label("M",(5,0),S); label("N",(8,0),S); label("13",(2.5,6),NW); label("13",(11.5,6),NE); label("13",(6.5,-1),S); label("3",(6.5,12),N); label("12",(5,6),W); label("12",(8,6),E); label("5",(2.5,0),N); label("5",(10.5,0),N); [/asy]$

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User:Annoyingpi: Difference between revisions

Latest revision as of 00:30, 10 December 2008

@@ Line 1: / Line 1: @@
-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>.
+<asy>
+dot((-1,0));
+dot((1,0));
+dot((0,1.5));
+dot((0,-1.5));
+dot((0.3,0.3));
-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>
+draw((1,0)--(0,1.5)--(-1,0)--(0,-1.5)--cycle);
-\frac{dy}{dx} = -x \frac1{\sqrt{4-x^2}}.
-</cmath>
-Therefore the integral is
+label("$P_1$",(-1,0),W);
-<cmath>
+label("$P_2$",(1,0),E);
-\boxed{\int_{-\sqrt3}^{\sqrt3} \sqrt{1+ \frac{x^2}{4-x^2}} \, dx}.
+label("$Q_1$",(0,1.5),N);
-</cmath>
+label("$Q_2$",(0,-1.5),S);
+label("$X$",(0.3,0.3),SW);
+</asy>
-'''IGNORE EVERYTHING BELOW:'''
 <asy>