<Text-field layout="Heading 1" style="Heading 1">Aufgabe 1</Text-field>

restart;

Int(rho, V) = Int(1/(x^2+y^2+z^2), V);

Int( Int( Int( 1/(x^2+y^2+z^2), z=0.1..5), y=0.1..5), x=0.1..5);

int( int( int( 1/(x^2+y^2+z^2), z=0.1..5), y=0.1..5), x=0.1..5);

M := %*8;

<Text-field layout="Heading 1" style="Heading 1">Aufgabe 2</Text-field>

restart;

f:=x -> piecewise(x>=0 and x<1,5,x>=1 and x<=4,5/x, x>4 and x<=5,5/4);

plot(f(x),x=0..5);

Int( Int( Int(r, z=0..f(r)), r=0..5), t=0..2*Pi );

int( int( int(r, z=0..f(r)), r=0..5), t=0..2*Pi );

<Text-field layout="Heading 1" style="Heading 1">Aufgabe 3</Text-field>

Kreis mit Radius NiMmJSJyRzYjIiIh und Mittelpunkt (R, 0) in (x,z)-Ebene: NiMvLCYqJCwmJSJ4RyIiIiUiUkchIiIiIiNGKCokJSJ6R0YrRigqJCYlInJHNiMiIiFGKw==

Substitution x -> r ergibt f\374r z(r): NiMvLSUiekc2IyUickctJSVzcXJ0RzYjLCYqJCZGJzYjIiIhIiIjIiIiKiQsJkYnRjElIlJHISIiRjBGNQ==

Bereich von z : NiQxLCQtJSVzcXJ0RzYjLCYqJCYlInJHNiMiIiEiIiMiIiIqJCwmRitGLyUiUkchIiJGLkYzRjMlInpHMUY0RiU=

Bereich von r : NiQxLCYlIlJHIiIiJiUickc2IyIiISEiIkYoMUYoLCZGJUYmRidGJg==

Integrationgebiet also: V = { ( NiUlInpHJSJyRyUidEc= ) \246 NiMxLCQtJSVzcXJ0RzYjLCYqJCYlInJHNiMiIiEiIiMiIiIqJCwmRitGLyUiUkchIiJGLkYzRjMlInpH , NiMxJSJ6Ry0lJXNxcnRHNiMsJiokJiUickc2IyIiISIiIyIiIiokLCZGK0YvJSJSRyEiIkYuRjM= , NiQxLCYlIlJHIiIiJiUickc2IyIiISEiIkYoMUYoLCZGJUYmRidGJg== , NiQxIiIhJSJ0RzFGJSomIiIjIiIiJSNQaUdGKQ== }

Resultat: NiMvJSJWRyoqIiIjIiIiKiQlI1BpR0YmRicmJSJyRzYjIiIhRiYlIlJHRic=

restart;

Int( Int( Int(r, z=-sqrt(r0^2-(r-R)^2)..sqrt(r0^2-(r-R)^2) ), r=R-r0..R+r0 ), t=0..2*Pi);

int( int( int(r, z=-sqrt(r0^2-(r-R)^2)..sqrt(r0^2-(r-R)^2) ), r=R-r0..R+r0 ), t=0..2*Pi);

<Text-field layout="Heading 1" style="Heading 1">Aufgabe 4</Text-field>

Gegeben: NiMvLSUkcmhvRzYjJSJyRyooJkYlNiMiIiEiIiIsJiUiUkdGLEYnISIiRixGLkYv

<Text-field layout="Heading 2" style="Heading 2">(a)</Text-field>

Gesucht: NiMvJSJNRy0lJEludEc2JComJSRyaG9HIiIiJSJWR0YqRis=

Kugelkoordiaten geeignet

restart; Digits := 6:

M = Int( Int( Int(rho[0]*(R-r)/R*r^2*cos(theta), r=0..R), theta=-Pi/2..Pi/2), t=0..2*Pi);

M = int( int( int(rho[0]*(R-r)/R*r^2*cos(theta), r=0..R), theta=-Pi/2..Pi/2), t=0..2*Pi);

<Text-field layout="Heading 2" style="Heading 2">(b, c)</Text-field>

Resultat: NiMvJiUkcmhvRzYjIiIhIiVJYw== NiMqJiUja2dHIiIiKiQlIm1HIiIkISIi

Vergleich: NiMvJiUkcmhvRzYjJSRhdmdHIiU1OQ== NiMqJiUja2dHIiIiKiQlIm1HIiIkISIi

Tats\344chlich: NiMvJiUkcmhvRzYjIiIhIiYrPyI= NiMqJiUja2dHIiIiKiQlIm1HIiIkISIi

mSun := 1.989*10^30: rSun := 6.96*10^8:

rho_0 := evalf( 3*mSun/(rSun^3*Pi) ); rho_avg := evalf( mSun/(4*Pi*rSun^3/3) );

<Text-field layout="Heading 1" style="Heading 1">Aufgabe 5</Text-field>

Gegeben: NiMvLSUieEc2IyUidEcqJiUiY0ciIiJGJ0Yq, NiMvLSUieUc2IyUidEcqJiUiY0ciIiJGJ0Yq, NiMvLSUiekc2IyUidEctJSVzcXJ0RzYjLCYtJSJ4R0YmIiIiLSUieUdGJkYu

Veranschaulichung:

restart; Digits := 6:

plot3d(sqrt(x+y), x=0..5, y=0..5 );

<Text-field layout="Heading 2" style="Heading 2">(a)</Text-field>

Int( Int( Int( Int(1, z=0..sqrt(x+y) ), y=0..c*t ), x=0..c*t ), t=0..T );

Vol := int( int( int( int(1, z=0..sqrt(x+y) ), y=0..c*t ), x=0..c*t ), t=0..T );

<Text-field layout="Heading 2" style="Heading 2">(b)</Text-field>

eval( Vol, {c=3*10^8, T=1.5*10^10*365*24*3600} );

Vol_0 := evalf(%);

<Text-field layout="Heading 2" style="Heading 2">(c)</Text-field>

R_Vol := evalf( ( (3*Vol_0)/(4*Pi) )^(1/3) );

R_ct := 3*10^8*1.5*10^10*365*24*3600;

R_Vol/R_ct;