设平面图形A由x^2+y^2=x确定,求该平面图形的面积及其绕直线x=2旋转一周所得的旋转体的体积,尽快啊……
设平面图形A由x^2+y^2=x确定,求该平面图形的面积及其绕直线x=2旋转一周所得的旋转体的体积,尽快啊……
设平面图形A由x^2+y^2<2x与y>=x确定,求该平面图形的面积
及其绕直线x=2旋转一周所得的旋转体的体积,尽快啊……
设平面图形A由x^2+y^2=x确定,求该平面图形的面积及其绕直线x=2旋转一周所得的旋转体的体积,尽快啊……
解法一(以x为积分变量求解):
∵(自己作图)x²+y²=2x与y=x的交点是(0,0)与(1,1)
∴所求面积=∫[√(2x-x²)-x]dx
=∫√(1-(x-1)²)dx-∫xdx
=∫cos²tdt-1/2 (在第一个积分中,令x-1=sint)
=∫[(1+cos(2t))/2]dt-1/2
=π/4-1/2
所求体积=∫2π(2-x)[√(2x-x²)-x]dx
=2π[∫(2-x)√(1-(x-1)²)dx-∫(2x-x²)dx
=2π[∫(1-sint)cos²tdt-(1-1/3)] (在第一个积分中,令x-1=sint)
=2π[∫(1/2+cos(2t)/2-sintcos²t)dt-2/3]
=2π[(1/3+π/4)-2/3]
=π²/2-2π/3
解法二(以y为积分变量求解):
∵(自己作图)x²+y²=2x与y=x的交点是(0,0)与(1,1)
∴所求面积=∫[y-(1-√(1-y²))]dy
=∫√(1-y²)dy+∫(y-1)dy
=∫cos²tdt+(1/2-1) (在第一个积分中,令y=sint)
=∫(1/2+cos(2t)/2)dt-1/2
=π/4-1/2
所求体积=∫π[(1+√(1-y²))²-(2-y)²]dy
=2π∫[√(1-y²)-(1-2y+y²)]dy
=2π[∫√(1-y²)dy-∫(1-2y+y²)dy]
=2π[∫cos²tdt-(1-1+1/3)] (在第一个积分中,令y=sint)
=2π[∫(1/2+cos(2t)/2)dt-1/3]
=2π(π/4-1/3)
=π²/2-2π/3
x^2+y^2=2x, y=x y=√(2x-x^2)
交于O(0,0) A(1,1)
S=∫[0,1][√(2x-x^2)-x]dx
=∫[0,1]√(1-(x-1)^2)d(x-1) -(1/2)
x-1=sinu x=0 u=-π/2, x=1,u=0
=∫[-π/2,0]cosu^2du-1/2
=(1/2...
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x^2+y^2=2x, y=x y=√(2x-x^2)
交于O(0,0) A(1,1)
S=∫[0,1][√(2x-x^2)-x]dx
=∫[0,1]√(1-(x-1)^2)d(x-1) -(1/2)
x-1=sinu x=0 u=-π/2, x=1,u=0
=∫[-π/2,0]cosu^2du-1/2
=(1/2)∫[-π/2,0](1+cos2u)du -1/2
=(1/2)*π/2-1/2
=π/4-1/2
V=∫[0,1] 2π*(2-x)dx∫[x,√(2x-x^2)]dy
=2π∫[0,1](2-x)*[√(2x-x^2)-x]dx
=2π*[(1/3)√(2x-x^2)^3 +(1/2)arcsin(x-1)+(1/2)(x-1)√(2x-x^2)-x^2+(1/3)x^3] |[0,1]
=2π*[(1/3)+(1/2)(π/2)-1+(1/3)]
=(1/2)π^2 -2π/3
∫(2-x)[√(2x-x^2)-x]dx=∫(1-x)√(1-(x-1)^2) +√(1-(x-1)^2)d(x-1)-x^2+(1/3)x^3
x-1=sinu =∫-sinucosu^2+cosu^2 du -x^2+(1/3)x^3
=(1/3)cosu^3+(1/2)u+(1/4)sin2u -x^2+(1/3)x^3
=(1/3)√(2x-x^2)^3+(1/2)arcsin(x-1)+(1/2)(x-1)√(2x-x^2)-x^2+(1/3)x^3
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将x^2+y^2<2x化为圆方程标准公式(x-1)^2+y^2<1
PS:自己画个草图
该曲线是以(1,0)为圆点,半径为1的圆
与y>=x确定的图形的面积为:¼圆面积—½×1×1=¼π-½
A面积为圆(x-1)²+y²=1被直线y=x截得劣弧面积
S=S扇形-S三角形=πr²/4-1*1/2=π/4-1/2
体积:扇形旋转所得体积 - 三角形旋转所得体积
扇形旋转所得体积为:圆环体积/4
三角形旋转所得体积为:圆锥1体积(大)-圆锥2体积(小)-圆柱体积
V=2π²r²*R/4-(πr1...
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A面积为圆(x-1)²+y²=1被直线y=x截得劣弧面积
S=S扇形-S三角形=πr²/4-1*1/2=π/4-1/2
体积:扇形旋转所得体积 - 三角形旋转所得体积
扇形旋转所得体积为:圆环体积/4
三角形旋转所得体积为:圆锥1体积(大)-圆锥2体积(小)-圆柱体积
V=2π²r²*R/4-(πr1²h1/3-πr2²h2/3-πr2²*(h1-h2))
=2π²*1²*1²/4-(π*2²*2/3-π*1²*1/3-π*1²*(2-1))
=π²/2-4π/3
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