fx在[0,1]上可导,满足xf'(x)=f(x)+3x^2,求fx使得y=f(x)与直线x=0,x=1,y=0所围成的
fx在[0,1]上可导,满足xf'(x)=f(x)+3x^2,求fx使得y=f(x)与直线x=0,x=1,y=0所围成的平面绕x轴一周后旋转体体积最小
赞 姜士安 幼苗
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xf'(x)=f(x)+3x^2
x*dy/dx=y+3x^2
dy/dx=y/x+3x
设u=y/x,y=ux,dy=udx+xdu,dy/dx=u+x*du/dx
u+x*du/dx=u+3x
x*du/dx=3x
du/dx=3
u=3x+C
y=3x^2+Cx
设所求旋转体体积为V
V=∫(0→1)π*[f(x)]^2*dx
=π∫(0→1)(3x^2+Cx)^2*dx
=π∫(0→1)(9x^4+6Cx^3+C^2*x^2)dx
=π*[(9/5)x^5+(3C/2)x^4+(C^2/3)*x^3]|(0→1)
=π(9/5+3C/2+C^2/3)
当dV/dC=0时,V具有最小值
dV/dC=π(3/2+2C/3)=0
C=-9/4
故f(x)=3x^2-9x/4
x*dy/dx=y+3x^2
dy/dx=y/x+3x
设u=y/x,y=ux,dy=udx+xdu,dy/dx=u+x*du/dx
u+x*du/dx=u+3x
x*du/dx=3x
du/dx=3
u=3x+C
y=3x^2+Cx
设所求旋转体体积为V
V=∫(0→1)π*[f(x)]^2*dx
=π∫(0→1)(3x^2+Cx)^2*dx
=π∫(0→1)(9x^4+6Cx^3+C^2*x^2)dx
=π*[(9/5)x^5+(3C/2)x^4+(C^2/3)*x^3]|(0→1)
=π(9/5+3C/2+C^2/3)
当dV/dC=0时,V具有最小值
dV/dC=π(3/2+2C/3)=0
C=-9/4
故f(x)=3x^2-9x/4
1年前
5
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