幫忙一下關於微積分，有幾題不會算，謝謝!

Question

我將題目弄對圖片放在下面網址
http://webhd.mcu.edu.tw/download/cant%20understand.GIF?para=bvR7H4BUjdINEJXTC9zMTEzNDE0MS9jYW50IHVuZGVyc3RhbmQuR0lGdllidlI3SDRCVw==

如果以上網址進不去，試下列網址
http://ima.csim.hk.edu.tw/g2hosts/main.php?g2_view=core.DownloadItem&g2_itemId=16307&g2_serialNumber=2

Answer 1

1.設最小距離為d,則d²=x²+y²+z² 且滿足xyz²=2令L= x²+y²+z²+λ(xyz²-2),則∂L/∂x=2x+λyz²∂L/∂y=2y+λxz²∂L/∂z=2z+2λxyz解聯立方程式2x+λyz²=0 ............➀2y+λxz²=0 ............➁2z+2λxyz=0 ..........➂xyz²=2 .....................➃由➂知z(1+λxy)=0, z=0 或 1+ λxy =0若z=0代入➀➁則x=0,y=0 ⇒ xyz²=0 與➃不合1+ λxy =0 ......➄ xy=-1/λ代入➃得z²=-2λ代入➀得2x-2λ²y=0, x=λ²y代入➁得2y+λ(λ²y)(-2λ)=0y-λ4y=0, y(1-λ4)=0, 若y=0代入 ➃0=2不合1-λ4=0, λ=±1, 若 λ =1由 ➄ 知xy=-1代入➃z²=-2不合λ =-1得z²=2代入➀2x+(-1)y(2)=0, 2x-2y=0, x=y代入➃x(x)(2)=2, x²=1, ∴y²=1d²=x²+y²+z²=1+1+2=4∴d=22.R={(x,y)∣0≦x≦10, 0≦y≦π}∫∫R ycos(xy) dA=∫0π∫010 ycos(xy) dxdy=∫0π sin(xy)∣x=010 dxdy=∫0π sin(10y) dy= -(1/10)cos(10y)∣0π= -(1/10)(cos(10π) -cos0)= -(1/10)(0)= 03.∫01∫x1 ey² dydx=∫01∫0y ey² dxdy=∫01 ey²∫0y 1 dxdy=∫01 yey² dy= (1/2)ey²∣01 = (e-1)/24.R={(x,y)∣x²+y²≦2²}∫∫R √(x²+y²) dA=∫02π∫02 √(r²) rdrdθ=∫02π∫02 r²drdθ=∫02π (1/3)r³∣02 dθ=∫02π (8/3) dθ=(16π)/35.E={(x,y,z)∣ x²+y²+z² ≦1}(x=ρsinφcosθy=ρsinφsinθz=ρcosφx²+y²+z²=ρ²dxdydz=ρ² sinφdρdφdθ)∫∫∫E ( x²+y²+z² ) dV=∫02π∫0π∫01 (ρ²)ρ² sinφdρdφdθ=∫02π∫0π sinφ∫01 ρ4 dρdφdθ=∫02π∫0π (sinφ)(1/5)[ρ5]01 dφdθ= (1/5)∫02π∫0π (sinφ) dφdθ= (1/5)∫02π [-cosφ]0π dθ= (1/5)∫02π [-(-1)-(-1)] dθ= (1/5)∫02π 2 dθ= 4π/5

2006-08-22 07:10:36 補充：
抱歉ㄚ,我又回來看了一下,第2題題目看錯了,是0~1不是0~10
=∫(0~π)∫(0~1) ycos(xy) dxdy
=∫(0~π) [sin(xy)](x=0~1) dy
=∫(0~π) siny dy
=[-cosy](0~π)
=-(-1)-(-1)
=2