blatt09.mw

Blatt 9 Polarkoordinaten und Dreifachintegrale

> restart:with(plots):with (student):

Warning, the name changecoords has been redefined

Vorlesungs-Beispiel zum Dreifachintegral

> x0:=0.1; y0:=0.15; z0:=x0+y0; z1:=1-z0;

x0 := .1

y0 := .15

z0 := .25

z1 := .75

> p1:=plot3d(x+y,x=0..0.5,y=0..0.5-x,grid=[10,10],axes=normal):

> p2:=plot3d(1-x-y,x=0..0.5,y=0..0.5-x,grid=[10,10],transparency=0.5):

> p3:=spacecurve([x,0.5-x,0],x=0..0.5,thickness=2,color=grey):

> Stelle:=spacecurve([x0+0.004*cos(t),y0+0.004*sin(t),0],t=0..2*Pi,thickness=2,color=blue):

> l:=spacecurve([x0,y0,z0*t+(1-t)*z1],t=0..1,thickness=2,color=blue):

> q:=plot3d([x0,u,(x0+u)*v+(1-v)*(1-x0-u)],u=0..(0.5-x0),v=0..1,grid=[10,10],color=green):

> q0:=spacecurve([x0,u,0],u=0..(0.5-x0),thickness=2,color=green):

> display(p1,p2,p3,Stelle,l,q,q0);

[Plot]

> I1:=Tripleint(1,z=(x+y)..(1-x-y),y=0..(1/2-x),x=0..1/2);

I1 := Int(Int(Int(1, z = x+y .. 1-x-y), y = 0 .. 1/2-x), x = 0 .. 1/2)

> value(I1);

Blatt 9 T 27

> p1:=plot3d((x^2+y^2)/4,x=-4..4,y=4-sqrt(16-x^2)..4+sqrt(16-x^2),color=red):

> p2:=plot3d([4*cos(v),4+4*sin(v),u], v=0..2*Pi,  u=0..(8+8*sin(v)), grid=[20,20], scaling=unconstrained, axes=normal, transparency=0.7):

> display(p1,p2);

[Plot]

> I1:=Tripleint(1,z=0..1/4*(x^2+y^2),y=4-sqrt(16-x^2)..4+sqrt(16-x^2),x=-4..4);

I1 := Int(Int(Int(1, z = 0 .. 1/4*x^2+1/4*y^2), y = 4-(16-x^2)^(1/2) .. 4+(16-x^2)^(1/2)), x = -4 .. 4)

> value(I1);

96*Pi

> int(1/3*(x^2+32)*sqrt(16-x^2),x);

-1/12*x*(16-x^2)^(3/2)+6*x*(16-x^2)^(1/2)+96*arcsin(1/4*x)

Blatt 9 T 28

> p1:=plot3d([cos(v),sin(v),u],v=0..2*Pi, u=-sin(v)..sin(v)):

> p2o:=plot3d(sqrt(1-x^2),x=-1..1,y=-sqrt(1-x^2)..sqrt(1-x^2), scaling=constrained):

> p2u:=plot3d(-sqrt(1-x^2),x=-1..1,y=-sqrt(1-x^2)..sqrt(1-x^2)):

> display(p1,p2o,p2u);

[Plot]

> I1:=8*Tripleint(1, z=0..sqrt(r^2-x^2), y=0..sqrt(r^2-x^2), x=0..r);

I1 := 8*Int(Int(Int(1, z = 0 .. (r^2-x^2)^(1/2)), y = 0 .. (r^2-x^2)^(1/2)), x = 0 .. r)

> value(I1);

16/3*r^3

Blatt 9 H 15

> f:=y/x^4; f:=unapply(f,(x,y)):

f := y/x^4

> fp:=f(r*cos(v),r*sin(v)); fp:=unapply(fp,(r,v)):

fp := sin(v)/(r^3*cos(v)^4)

> px:=spacecurve([t,0,0], t=0..2.5):

> py:=spacecurve([0,t,0], t=0..2.5):

> pr1:=plot3d([cos(v),sin(v),u], v=0..Pi/4, u=0..fp(1,v)):

> pr2:=plot3d([2*cos(v),2*sin(v),u], v=0..Pi/4, u=0..fp(2,v)):

> pr:=plot3d([r*cos(Pi/4),r*sin(Pi/4), u], r=1..2, u=0..fp(r,Pi/4)):

> pf:=plot3d([r*cos(v),r*sin(v),fp(r,v)],r=1..2,v=0..Pi/4,scaling=constrained, axes=normal, color=red):

> display(px,py,pr1,pr2,pr,pf);

[Plot]

> F1:=Doubleint(fp(r,v)*r,r=1..2,v=0..Pi/4);

F1 := Int(Int(sin(v)/(r^2*cos(v)^4), r = 1 .. 2), v = 0 .. 1/4*Pi)

> value(F1);

1/3*2^(1/2)-1/6

Blatt 9 E 9

> pfo:=plot3d([r*cos(v),r*sin(v),sqrt(4-r^2)],v=-Pi/2..Pi/2,r=2*cos(v)..2, scaling=constrained, color=blue):

> pfu:=plot3d([r*cos(v),r*sin(v),-sqrt(4-r^2)],v=-Pi/2..Pi/2,r=2*cos(v)..2, color=blue):

> pfo1:=plot3d([r*cos(v),r*sin(v),sqrt(4-r^2)],v=Pi/2..3*Pi/2,r=0..2,  color=blue):

> pfu1:=plot3d([r*cos(v),r*sin(v),-sqrt(4-r^2)],v=Pi/2..3*Pi/2,r=0..2,  color=blue):

> ps:=plot3d([2*cos(v)*cos(v),2*cos(v)*sin(v),2*u*sqrt(1-cos(v)^2)],v=-Pi/2..Pi/2,u=-1..1, axes=normal):

> display(pfo,pfu,pfo1,pfu1,ps,tickmarks=[1,1,1]);

[Plot]

> F2:=2*Doubleint(sqrt(4-r^2)*r,r=0..2*cos(v),v=-Pi/2..Pi/2);

F2 := 2*Int(Int((4-r^2)^(1/2)*r, r = 0 .. 2*cos(v)), v = -1/2*Pi .. 1/2*Pi)

> value(F2);

16/3*Pi-64/9

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