求曲线绕指定坐标旋转体的体积.1、y=2x,x=1,y=0,分别绕X和y轴;2、Y=sinx（0,π）与y=0,绕x轴.还有这两题：求曲线所围成的圆形面积。1、y=2-x^2,y=x2、y=-x,y=2x与x=2

Question

求曲线绕指定坐标旋转体的体积.1、y=2x,x=1,y=0,分别绕X和y轴;2、Y=sinx（0,π）与y=0,绕x轴.
还有这两题：求曲线所围成的圆形面积。1、y=2-x^2,y=x 2、y=-x,y=2x与x=2

Answer 1

1、y=2x,x=1,y=0
绕x轴
V=πʃ(0~1)4x²dx
=π*4/3x³|(0~1)
=4π/3
绕y轴
V=πʃ(0,2)(1-1/2y)²dy
=πʃ(0,2)(1-y+y²/4)dy
=π(y-y²/2+y³/12)|(0~2)
=(2-2+2/3)π
=2π/3
2、Y=sinx（0,π）与y=0,绕x轴.
V=πʃ(0~π)(sinx)^2dx
=π/2ʃ(0~π)(1-cos2x)dx
=π/2(x-1/2sin2x)|(0~π)
=π/2(π-0)
=π²/2
面积：
1、y=2-x^2,y=x
{y=2-x^2
{y=x
==>交点A(-2,-2),B(1,1）
S=ʃ（-2~1）（2-x^2-x)dx
=(2x-1/3x^3-1/2x^2)|(-2~1)
=(2-1/3-1/2)-(-4+8/3-2)
=6-5/6+2-8/3
=8-7/2
=9/2
2、y=-x,y=2x与x=2
S=ʃ(0~2)(2x+x)dx
=3/2x^2|(0~2)
=6