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  • area of plane region | planetmath.org
    P P P Then the area of the region equals to the path integral A 1 2 P x d y y d x A 1 2 subscript contour integral P x d y y d x displaystyle A frac 1 2 oint P x dy y dx 1 taken in the positive i e anticlockwise circling direction Remarks 1 The formula 1 can be gotten as a special case of Green s theorem by setting F 1 2 y x assign normal F 1 2 y x vec F frac 1 2 y x 2 Because x d y y d x d x y x d y y d x d x y x dy y dx d xy we have 0 1 2 P x d y y d x 0 1 2 subscript contour integral P x d y y d x 0 frac 1 2 oint P x dy y dx This equation may be added to or subtracted from 1 giving the alternative forms A P x d y P y d x A subscript contour integral P x d y subscript contour integral P y d x displaystyle A oint P x dy oint P y dx 2 3 The formulae 1 and 2 contain all other formulae concerning the planar area computing e g A a b f x d x A superscript subscript a b f x d x A int a b f x dx A 1 2 φ 1 φ 2 r φ 2 d φ A 1 2 superscript subscript subscript φ 1 subscript φ 2 superscript r φ 2 d φ A frac 1 2 int varphi 1 varphi 2 r varphi 2 d varphi the former of which is factually same as the latter form of

    Original URL path: http://www.planetmath.org/areaofplaneregion (2016-04-25)
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  • area of polygon | planetmath.org
    all modules sparql sparql module User error Socket error Could not connect to http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A26A42 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D proxy 0 Connection refused in ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in getFormat via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A26A42 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit area of polygon Let the vertices of a planar polygon be x 1 y 1 x 2 y 2 x n y n subscript

    Original URL path: http://www.planetmath.org/areaofpolygon (2016-04-25)
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  • area of spherical calotte by means of chord | planetmath.org
    sparql module User error Socket error Could not connect to http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A51M25 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D proxy 0 Connection refused in ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in getFormat via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A51M25 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit area of spherical calotte by means of chord Let the arc P R P R PR of a circle with radius r r r rotate about the

    Original URL path: http://www.planetmath.org/areaofsphericalcalottebymeansofchord (2016-04-25)
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  • area of spherical zone | planetmath.org
    request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit area of spherical zone Let us consider the circle x r 2 y 2 r 2 superscript x r 2 superscript y 2 superscript r 2 x r 2 y 2 r 2 with radius r r r and centre r 0 r 0 r 0 A spherical zone may be thought to be formed when an arc of the circle rotates around the x x x axis For finding the are of the zone we can use the formula A 2 π a b y 1 d y d x 2 d x A 2 π superscript subscript a b y 1 superscript d y d x 2 d x displaystyle A 2 pi int a b y sqrt 1 left frac dy dx right 2 dx 1 of the entry area of surface of revolution Let the ends of the arc correspond the values a a a and b b b of the abscissa such that b a h b a h b a h is the height of the spherical zone In the formula we must use the solved form y r x x 2 y plus or minus r x superscript x 2 y pm sqrt rx x 2 of the equation of the circle The formula then yields A 2 π a b r x x 2 1 r x r x x 2 2 d x 2 π a b r d x 2 π r b a A 2 π superscript subscript a b r x superscript x 2 1 superscript r x r x superscript x 2 2 d x 2 π superscript subscript a b r d x 2

    Original URL path: http://www.planetmath.org/areaofsphericalzone (2016-04-25)
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  • area under Gaussian curve | planetmath.org
    Connection refused in ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in getFormat via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A51M04 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit area under Gaussian curve Theorem The area between the curve y e x 2 y superscript e superscript x 2 y e x 2 and the x x x axis equals π π sqrt pi i e e x 2 d x π superscript subscript superscript e superscript x 2 d x π int infty infty e x 2 dx sqrt pi Proof The square of the area is e x 2 d x 2 superscript superscript subscript superscript e superscript x 2 d x 2 displaystyle bigg int infty infty e x 2 dx bigg 2 lim a a a e x 2 d x 2 absent subscript normal a superscript superscript subscript a a superscript e superscript x 2 d x 2 displaystyle lim a to infty bigg int a a e x 2 dx bigg 2 lim a a a e x 2 d x a a e y 2 d y absent subscript normal a superscript subscript a a normal superscript e superscript x 2 d x superscript subscript a a superscript e superscript y 2 d y displaystyle lim a to infty int a a e x 2 dx cdot int a a e y 2 dy lim a a a a a e x 2 y 2 d x d y absent subscript normal a superscript subscript a a superscript subscript a a superscript e superscript x 2 superscript y 2 d x d y displaystyle lim a to infty int a a int a a e x 2 y 2 dx dy lim R 0 R 0 2 π e r 2 r d r d φ absent subscript normal R superscript subscript 0 R superscript subscript 0 2 π superscript e superscript r 2 r d r d φ displaystyle lim R to infty int 0 R int 0 2 pi e r 2 r dr d varphi lim R 2 π 0 R e r 2 r d r absent subscript normal

    Original URL path: http://www.planetmath.org/areaundergaussiancurve (2016-04-25)
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  • argument of product and quotient | planetmath.org
    φ 2 i subscript φ 2 subscript φ 1 subscript φ 2 subscript φ 1 subscript φ 2 i subscript φ 1 subscript φ 2 subscript φ 1 subscript φ 2 cos varphi 1 i sin varphi 1 cos varphi 2 i sin varphi 2 cos varphi 1 cos varphi 2 sin varphi 1 sin varphi 2 i sin varphi 1 cos varphi 2 cos varphi 1 sin varphi 2 Using the addition formulas of cosine and sine we still obtain the formula cos φ 1 i sin φ 1 cos φ 2 i sin φ 2 cos φ 1 φ 2 i sin φ 1 φ 2 subscript φ 1 i subscript φ 1 subscript φ 2 i subscript φ 2 subscript φ 1 subscript φ 2 i subscript φ 1 subscript φ 2 displaystyle cos varphi 1 i sin varphi 1 cos varphi 2 i sin varphi 2 cos varphi 1 varphi 2 i sin varphi 1 varphi 2 1 The inverse number of cos φ 2 i sin φ 2 subscript φ 2 i subscript φ 2 cos varphi 2 i sin varphi 2 is calculated as follows 1 cos φ 2 i sin φ 2 cos φ 2 i sin φ 2 cos φ 2 i sin φ 2 cos φ 2 i sin φ 2 cos φ 2 i sin φ 2 cos 2 φ 2 sin 2 φ 2 1 subscript φ 2 i subscript φ 2 subscript φ 2 i subscript φ 2 subscript φ 2 i subscript φ 2 subscript φ 2 i subscript φ 2 subscript φ 2 i subscript φ 2 superscript 2 subscript φ 2 superscript 2 subscript φ 2 frac 1 cos varphi 2 i sin varphi 2 frac cos varphi 2 i sin varphi 2 cos varphi 2 i sin varphi 2 cos varphi 2 i sin varphi 2 frac cos varphi 2 i sin varphi 2 cos 2 varphi 2 sin 2 varphi 2 This equals cos φ 2 i sin φ 2 subscript φ 2 i subscript φ 2 cos varphi 2 i sin varphi 2 and since the cosine is an even and the sine an odd function we have 1 cos φ 2 i sin φ 2 cos φ 2 i sin φ 2 1 subscript φ 2 i subscript φ 2 subscript φ 2 i subscript φ 2 displaystyle frac 1 cos varphi 2 i sin varphi 2 cos varphi 2 i sin varphi 2 2 The equations 1 and 2 imply cos φ 1 i sin φ 1 cos φ 2 i sin φ 2 cos φ 1 i sin φ 1 cos φ 2 i sin φ 2 cos φ 1 φ 2 i sin φ 1 φ 2 subscript φ 1 i subscript φ 1 subscript φ 2 i subscript φ 2 subscript φ 1 i subscript φ 1 subscript φ 2 i subscript φ 2 subscript φ 1 subscript φ 2 i subscript φ 1 subscript φ 2

    Original URL path: http://www.planetmath.org/argumentofproductandquotient (2016-04-25)
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  • associativity of multiplication | planetmath.org
    3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A26A09 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D proxy 0 Connection refused in ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in getFormat via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A26A09 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit associativity of multiplication It s important to know the following interpretation of the associative law a b c a b c normal a normal b c normal normal a b c displaystyle a cdot b cdot c a cdot b cdot c 1 of multiplication in arithmetics and elementary algebra A product b c normal b c b

    Original URL path: http://www.planetmath.org/associativityofmultiplication (2016-04-25)
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  • asymptote | planetmath.org
    Socket error Could not connect to http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A00A35 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D proxy 0 Connection refused in ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in getFormat via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A00A35 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit asymptote If a plane curve γ γ gamma has a branch continuing infinitely far from the origin O O O then γ γ gamma may have an asymptote The direct line

    Original URL path: http://www.planetmath.org/asymptote (2016-04-25)
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