Архив разработки (5 Кб, Maple 7)
задача 1 ~ задача 2 ~ задача 3 ~ задача 4 ~ задача 5
ЗАДАЧА 1.
Преобразовать двойной интеграл в двукратный и расставит пределы по заданной области интгрирования D. Изменить порядок интегрирования. Перейти к полярным координатам.
Область D : , 0 .
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restart:with(plottools):with(plots): |
Warning, the names arrow and changecoords have been redefined
> |
j:=0:k:=0:
W1:=plot({sqrt(x),-1},x=0..3,y=-1..2,color=GREEN,thickness=2):
W2:=circle([0,0],sqrt(2),thickness=2):
for i from 0 to 3 by .1 do
w[j]:=line([i,-1],[i,sqrt(i)],color=green):j:=j+1:od:
for l from -1 to evalf(sqrt(2)) by evalf(.1) do
wc[k]:=line([0,l],[sqrt(abs(2-l^2)),l],color=BLACK):k:=k+1:od: |
> |
plots[display]([W1,W2,w[e]$e=0..j-1,wc[r]$r=0..k-1],scaling=constrained); |
![[Maple Plot]](integrals/%C3%8A%C3%B0%C3%A0%C3%B2%C3%AD%C3%BB%C3%A5%20%C3%A8%C3%AD%C3%B2%C3%A5%C3%A3%C3%A0%C3%AB%C3%BB13.gif)
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print(`Изменим порядок интегрирования:`);
Int(``,x=0..1)*Int(f(x,y),y=-1..sqrt(x))+
Int(``,x=1..sqrt(2))*Int(f(x,y),y=-(-x^2+2)^(1/2)..(-x^2+2)^(1/2))=
Int(``,y=-1..0)*Int(f(x,y),x=0..sqrt(2-y^2))+
Int(``,y=0..1)*int(f(x,y),x=y^2..sqrt(2-y^2)); |


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print(`Преобразование в полярный координаты:`);
Int(``,phi=-Pi/2..-Pi/4)*Int(rho*f(rho,phi),rho=0..-1/sin(phi))+
Int(``,phi=-Pi/4..Pi/4)*Int(rho*f(rho,phi),rho=0..sqrt(2))+
Int(``,phi= Pi/4.. Pi/2)*Int(rho*f(rho,phi),rho=0..cos(phi)/sin(phi)^2);; |



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ЗАДАЧА 2.
Вычислить объём тела, ограниченного данными поверхностями, с помощью двойного интеграла.

Warning, the names arrow and changecoords have been redefined
> |
plot3d({x^2+y^2},x=0..1,y=x..2*x,style=wireframe,filled=true,style=HIDDEN,scaling=unconstrained,axes=normal,orientation=[-44,53],color=blue);
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![[Maple Plot]](integrals/%C3%8A%C3%B0%C3%A0%C3%B2%C3%AD%C3%BB%C3%A5%20%C3%A8%C3%AD%C3%B2%C3%A5%C3%A3%C3%A0%C3%AB%C3%BB19.gif)
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V=Doubleint(x^2+y^2,y=x..2*x,x=0..1),V=Int(int(x^2+y^2,y=x..2*x),x=0..1);
V=int(int(x^2+y^2,y=x..2*x),x=0..1);
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


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ЗАДАЧА 3.
Вычислить объём тела, ограниченного данными поверхностями, с помощью тройного интеграла.

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restart:with(plottools):with(plots):with(student): |
Warning, the names arrow and changecoords have been redefined
> |
Q1:=cylinder([0,0,0],3,.01):
Q2:=cylinderplot(3,theta=0..2*Pi,z=0..10,style=wireframe,linestyle=DOT,color=BLACK):
Q3:=cylinderplot(2,theta=0..2*Pi,z=0..10,style=patchnogrid,lightmodel=light2):
Q4:=plot3d(10-x^2,x=-sqrt(10)..sqrt(10),y=-sqrt(10)..sqrt(10),style=patchnogrid):
plots[display]([Q1,Q2,Q3,Q4],axes=normal,orientation=[116,63]); |
![[Maple Plot]](integrals/%C3%8A%C3%B0%C3%A0%C3%B2%C3%AD%C3%BB%C3%A5%20%C3%A8%C3%AD%C3%B2%C3%A5%C3%A3%C3%A0%C3%AB%C3%BB23.gif)
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print(`Переходим в цилиндрическую систему координат:`);
assign(x = u*cos(v),
y = u*sin(v),
z = w);
'x' = u*cos(v);
'y' = u*sin(v);
'z' = w;
print(`Наше уравнение примет вид:`);
z:=10-x^2;
print(`Якобиан:`);
I=u; |








> |
V=4*Tripleint(u,w=0..10-x^2,u=2..3,v=0..Pi/2),
V=value(4*Tripleint(u,w=0..10-x^2,u=2..3,v=0..Pi/2));
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

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ЗАДАЧА 4.
Найти массу тела G, заданного в пространстве неравенствами и имеющего плотность .
G : 9 ; : 
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P1:=sphereplot(3,theta=0..2*Pi,phi=0..Pi):
P2:=sphereplot(6,theta=0..2*Pi-Pi/3,phi=0..Pi,lightmodel='light3'):
plots[display]([P1,P2],scaling=constrained,style=patch,axes=FRAME,orientation=[-13,57]);
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![[Maple Plot]](integrals/%C3%8A%C3%B0%C3%A0%C3%B2%C3%AD%C3%BB%C3%A5%20%C3%A8%C3%AD%C3%B2%C3%A5%C3%A3%C3%A0%C3%AB%C3%BB37.gif)
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print(`Перейдём в сферическую систему координат:`);
x:=rho*cos(phi)*sin(Theta);
y:=rho*sin(phi)*sin(Theta);
z:=rho*cos(Theta); |




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print(`Плотность нашего тела`); |
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mu:='1/sqrt(x^2+y^2+z^2)';
print(`Якобиан:`);I=rho^2*cos(Theta); |




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`Плотность нашего тела`; |
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print(`Собственно, масса:`);
M=Tripleint(rho^2*cos(Theta)*1/sqrt(x^2+y^2+z^2),rho=3..6,phi=0..2*Pi,Theta=-Pi/2..Pi/2);
M=Tripleint(rho^2*cos(Theta)*simplify(abs(1/sqrt(x^2+y^2+z^2))),rho=3..6,phi=0..2*Pi,Theta=-Pi/2..Pi/2);
#value(%);
M=Int(Int(int(rho*cos(Theta),rho = 3 .. 6),phi = 0 .. 2*Pi),Theta = -1/2*Pi .. 1/2*Pi),
M=Int(int(int(rho*cos(Theta),rho = 3 .. 6),phi = 0 .. 2*Pi),Theta = -1/2*Pi .. 1/2*Pi),
M=int(int(int(rho*cos(Theta),rho = 3 .. 6),phi = 0 .. 2*Pi),Theta = -1/2*Pi .. 1/2*Pi); |





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ЗАДАЧА 5.
Найти моменты инерции тела G относительно оси ОY, имеющего плотность .
G : ( ) ; 
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restart:with(student):with(plottools):with(plots): |
> |
W1:=plot3d({4-x^2-y^2},x=0..2,y=-sqrt(4-x^2)..0,style=PATCHCONTOUR,contours=7): |
> |
i:=0:k:=0:
for j from 1 to 4 by .5 do
W2[i]:=line([0,0,j],[sqrt(4-j),0,j],color=BLACK,thickness=1):i:=i+1:od:
for j from 1 to 4 by .5 do
W2[i]:=line([0,0,j],[0,-sqrt(4-j),j],color=BLACK,thickness=1):i:=i+1:od:
W3:=plot3d(1,x=0..2,y=-2..0,style=patchnogrid,color=magenta):
plots[display]([W1,W2[l]$l=0..i-1,W3],axes=normal,orientation=[28,66],style=contour); |
![[Maple Plot]](integrals/%C3%8A%C3%B0%C3%A0%C3%B2%C3%AD%C3%BB%C3%A5%20%C3%A8%C3%AD%C3%B2%C3%A5%C3%A3%C3%A0%C3%AB%C3%BB54.gif)
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w=changecoords(4-x^2-y^2,[x,y,z],cylindrical,[u,v,w]);
simplify(%);allvalues(isolate(simplify(%),u)); |



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I['y']=Tripleint(u*changecoords(x^2+z^2,[x,y,z],cylindrical,[u,v,w]),u=0..(-w+4)^(1/2),v=-Pi/2..0,w=1..4);value(%);
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![I[y] = Int(Int(Int(u*(u^2*cos(v)^2+w^2),u = 0 .. (-w+4)^(1/2)),v = -1/2*Pi .. 0),w = 1 .. 4)](integrals/%C3%8A%C3%B0%C3%A0%C3%B2%C3%AD%C3%BB%C3%A5%20%C3%A8%C3%AD%C3%B2%C3%A5%C3%A3%C3%A0%C3%AB%C3%BB58.gif)
![I[y] = 45/8*Pi](integrals/%C3%8A%C3%B0%C3%A0%C3%B2%C3%AD%C3%BB%C3%A5%20%C3%A8%C3%AD%C3%B2%C3%A5%C3%A3%C3%A0%C3%AB%C3%BB59.gif)

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