直接利用積分求球體表面面積
2012-10-29 12:51 am
有無可以提供一項證明是直接利用圓形在坐標平面上的公式運用積分來求出其表面面積，本人在網上看見很多直接求體積再求表面面積，但未見直接求表面面積。如是不能求得，請提供理由。
回答 (5)
2012-10-29 6:35 pm
✔ 最佳答案
http://www.youtube.com/watch?v=Ej3PtDU57NUx^2+y^2=r^2
2x+2y dy/dx = 0
dy/dx = -x/y
Surface area
= in(-r to r) 2 pi y sqrt( 1+ (dy/dx)^2 ) dy
= in(-r to r) 2 pi y sqrt( 1+ x^2/y^2 ) dy
= in(-r to r) 2 pi y sqrt( (x^2+y^2) /y^2 ) dy
= in(-r to r) 2 pi y sqrt( r^2 /y^2 ) dy
= in(-r to r) 2 pi y sqrt( r^2 /y^2 ) dy
= in(-r to r) 2 pi sqrt( r^2 ) dy
= in(-r to r) 2 pi r dy
= 2 pi [ry] (-r to r)
= 2 pi (r^2 +r^2)
= 4 pi r^2
參考: maeducation.edu.hk
2012-11-14 9:04 am
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2012-11-11 10:05 pm
Surface area
= in(-r to r) 2 pi y sqrt( 1+ (dy/dx)^2 ) dy
= in(-r to r) 2 pi y sqrt( 1+ x^2/y^2 ) dy
= in(-r to r) 2 pi y sqrt( (x^2+y^2) /y^2 ) dy
= in(-r to r) 2 pi y sqrt( r^2 /y^2 ) dy
= in(-r to r) 2 pi y sqrt( r^2 /y^2 ) dy
= in(-r to r) 2 pi sqrt( r^2 ) dy
= in(-r to r) 2 pi r dy
= 2 pi [ry] (-r to r)
= 2 pi (r^2 +r^2)
= 4 pi r^2
because
x^2+y^2=r^2
2x+2y dy/dx = 0
dy/dx = -x/y
= in(-r to r) 2 pi y sqrt( 1+ (dy/dx)^2 ) dy
= in(-r to r) 2 pi y sqrt( 1+ x^2/y^2 ) dy
= in(-r to r) 2 pi y sqrt( (x^2+y^2) /y^2 ) dy
= in(-r to r) 2 pi y sqrt( r^2 /y^2 ) dy
= in(-r to r) 2 pi y sqrt( r^2 /y^2 ) dy
= in(-r to r) 2 pi sqrt( r^2 ) dy
= in(-r to r) 2 pi r dy
= 2 pi [ry] (-r to r)
= 2 pi (r^2 +r^2)
= 4 pi r^2
because
x^2+y^2=r^2
2x+2y dy/dx = 0
dy/dx = -x/y
2012-10-29 4:16 am
I'm asking to proof 4pi r^2 directly by using integration. could you please show me? the equation is x^2 + y^2 = r^2, plz help proof it.
2012-10-29 1:43 am
Not hard.
Just you need to know is this:
Surface area of curve y = f(x) rotate around x - axis is
2pi integrate(y sqrt(1 + dy/dx^2)) dx OR
2pi integrate(y sqrt(1 + dx/dy^2)) dy
2012-10-28 21:18:42 補充：
Required area, i.e. 4 pi r^2
= surface area generated by rotating the curve x^2 + y^2 = r^2 around x - axis.
Just you need to know is this:
Surface area of curve y = f(x) rotate around x - axis is
2pi integrate(y sqrt(1 + dy/dx^2)) dx OR
2pi integrate(y sqrt(1 + dx/dy^2)) dy
2012-10-28 21:18:42 補充：
Required area, i.e. 4 pi r^2
= surface area generated by rotating the curve x^2 + y^2 = r^2 around x - axis.
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原文連結 [永久失效]:
https://hk.answers.yahoo.com/question/index?qid=20121028000051KK00460