with(plots): R:=1:
zhumian1:=plot3d([R*cos(t),R*sin(t),z],z=-2*R..2*R, t=0..2*Pi,color=yellow):
zhumian2:=plot3d([R*cos(t),y,R*sin(t)],y=-2*R..2*R, t=0..2*Pi,color=green):
zhumian3:=plot3d([x,R*cos(t),R*sin(t)],x=-2*R..2*R, t=0..2*Pi,color=red):
display(zhumian1,zhumian2,zhumian3,scaling=constrained,orientation=[40,65]);
三个圆柱面所围成的立体：
with(plots): R:=1:
zhumian_a1:=plot3d([R*cos(t),R*sin(t),z],z=-R*sin(t)..R*sin(t),t=-Pi/4..Pi/4,color=yellow):
zhumian_a2:=plot3d([R*cos(t),R*sin(t),z],z=-R*cos(t)..R*cos(t),t=Pi/4..3*Pi/4,color=yellow):
zhumian_a3:=plot3d([R*cos(t),R*sin(t),z],z=-R*sin(t)..R*sin(t),t=3*Pi/4..5*Pi/4,color=yellow):
zhumian_a4:=plot3d([R*cos(t),R*sin(t),z],z=-R*cos(t)..R*cos(t),t=5*Pi/4..7*Pi/4,color=yellow):
zhumian_b1:=plot3d([R*cos(t),y,R*sin(t)],y=-R*sin(t)..R*sin(t),t=-Pi/4..Pi/4,color=green):
zhumian_b2:=plot3d([R*cos(t),y,R*sin(t)],y=-R*cos(t)..R*cos(t),t=Pi/4..3*Pi/4,color=green):
zhumian_b3:=plot3d([R*cos(t),y,R*sin(t)],y=-R*sin(t)..R*sin(t),t=3*Pi/4..5*Pi/4,color=green):
zhumian_b4:=plot3d([R*cos(t),y,R*sin(t)],y=-R*cos(t)..R*cos(t),t=5*Pi/4..7*Pi/4,color=green):
zhumian_c1:=plot3d([x,R*cos(t),R*sin(t)],x=-R*sin(t)..R*sin(t),t=-Pi/4..Pi/4,color=red):
zhumian_c2:=plot3d([x,R*cos(t),R*sin(t)],x=-R*cos(t)..R*cos(t),t=Pi/4..3*Pi/4,color=red):
zhumian_c3:=plot3d([x,R*cos(t),R*sin(t)],x=-R*sin(t)..R*sin(t),t=3*Pi/4..5*Pi/4,color=red):
zhumian_c4:=plot3d([x,R*cos(t),R*sin(t)],x=-R*cos(t)..R*cos(t),t=5*Pi/4..7*Pi/4,color=red):
a:=display(zhumian_a1,zhumian_a2,zhumian_a3,zhumian_a4):
b:=display(zhumian_b1,zhumian_b2,zhumian_b3,zhumian_b4):
c:=display(zhumian_c1,zhumian_c2,zhumian_c3,zhumian_c4):
display(a,b,c,scaling=constrained,orientation=[40,65],style=patch);
立体的体积是：
推导如下：
用Mathematica求体积（利用Boole函数）：
Integrate[Boole[x^2 + y^2 < R^2 && x^2 + z^2 < R^2 && y^2 + z^2 < R^2], {x, -Infinity, Infinity}, {y, -Infinity, Infinity}, {z, -Infinity, Infinity}]
选择R>0的结果：
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