Let us remind some well known facts.
Example 3. Plot the surface given by the following formulae:
where
Solution. A part of code:
Obj:=plot::Surface( [3*cos(u),3*sin(u),z], u=0..2*PI,z=0..10 ): plot(Obj,Scaling=Constrained);
Remark. The reader will solve this problem with the aid of the following MuPAD procedure:
plot::cylindrical
Problem 7
Compute the integral
over a region V bounded by the surfaces
Solution
First we will show region of integration. Here is MuPAD code generating model of the surface.
Obj1:=plot::Surface(
[2*cos(u),2*sin(u),z], Filled=FALSE,
u=0..2*PI,z=0..2
):
Obj2:=plot::Surface( [z*cos(u),z*sin(u),z], u=0..2*PI,z=0..2 ):
Obj3:=plot::Surface( [r*cos(u),r*sin(u),0], u=0..2*PI,r=0..2 ):
plot(Obj1, Obj2,Obj3)
The reader will complete the solution proceeding in a way shown in the preceding section by calculating Jacobian determinant and computing the integral.
Problem 8
Compute an integral
over a region V bounded by the surfaces
Solution
First let us calculate region of integration. The following code produces a model of the surface:
Obj1:=plot::Surface(
[r*4*cos(u)^2,r*4*cos(u)*sin(u),r*(4*cos(u))^2],
u=0..2*PI,r=0..1,
Mesh=[60,60]
):
Obj2:=plot::Surface(
[r*cos(u),r*sin(u),r^2],
u=0..2*PI,r=0..4,
VMesh =80,
Filled = FALSE,
VLinesVisible=FALSE
):
plot(Obj1,Obj2)
The remaining calculations we will leave to the readers.
Problem 9.
Compute the following integral
over the region V bounded by the surfaces
and including a point (0, 0, R). Final solution we will leave to the readers.
Model of a surface for R=3.
All interactive images in this article were produced with MuPAD and JavaView by M.Majewski
