mathPAD Online

Let us remind some well known facts.

Problem 10.

Compute an integral

for the region V bounded with surfaces

Solution

First let us visualize region of integration. Here is the code generating model of the surface:

Obj1:=plot::Surface(
     [t*s,(1-t)*s,1-s],
     s=0..1,t=0..1, 
     Mesh=[20,20]
):

Obj2:=plot::Surface(
   [t*s,(1-t)*s,0],
   s=0..1,t=0..1, Filled=FALSE, Mesh=[20,20]
):
   
Obj3:=plot::Surface(
   [0,(1-t)*s,1-s],
   s=0..1,t=0..1, 
   Filled=FALSE, 
   Mesh=[20,20]
):

Obj4:=plot::Surface(
   [t*s,0,1-s],
   s=0..1,t=0..1, 
   Filled=FALSE, 
   Mesh=[20,20]
):
    
plot(Obj1, Obj2, Obj3, Obj4, Scaling=Constrained);

Let us calculate Jacobian determinant

MA:=matrix([
 [diff(r*(1-t),r), diff(r*(1-t),t), diff(r*(1-t),s)],
 [diff(r*t*(1-s),r), diff(r*t*(1-s),t), diff(r*t*(1-s),s)],
 [diff(r*t*s,r), diff(r*t*s,t), diff(r*t*s,s)]
]):
linalg::det(MA)

In next step we will calculate the final integral.

simplify(
   subs(1/(x+y+z+1)^3, x=r*(1-t),y=r*t*(1-s),z=r*t*s)
);

Applying the theorem about substitution of variables and earlier results we compute the integral:

int(int(int(r^2*t/(r+1)^3,r=0..1),t=0..1),s=0..1); 

Problem 11.

Compute volume of the region bounded by the surfaces:

Solution we leave to the reader.

Summary

MuPAD software is a powerful tool in teaching mathematics. There are a number of benefits of using this software in calculus classes. Below we mention some of them.

• The teacher can use practical and exposing methods more intensively.
• Student has a chance for better and deeper understanding of mathematical notions.
• Student is able to create simple animations by himself and use MuPAD for complicated numeric computations.
• MuPAD provokes students to formulate new problems.

In the paper we only focused on a few possibilities of MuPAD within visualization of three-dimensional objects. Clearly, MuPAD offers many more powerful tools (e.g. plot::spherical or plot::cylindrical) that can be applied by the students in solving the problems presented in this article.

Bibliography

1. P. J. Davis, R. Hersh, Swiat matematyki, t.2 (Polish) [The Mathematical Experience], PWN, Warszawa 1994.
2. B. P. Demidovich, Zadachi i uprazhneniya po matematicheskomu analizu (Russian), Nauka, Moskva 1972.
3. G. M. Fichtenholz, Rachunek różniczkowy i całkowy, t. III (Polish) [A Course of Differential and Integral Calculus Vol. 3], PWN, Warszawa 1978.
4. M.Gewert, Z. Skoczylas, Analiza matematyczna 2 (Polish) [Mathematical Analysis], Oficyna Wydawnicza GiS, Wrocław 2002.
5. L. Górniewicz, R. S. Ingarden, Analiza matematyczna dla fizyków, t.2 (Polish) [Mathematical Analysis for Physicists], PWN, Warszawa 1985.
6. M.Majewski, MuPAD Pro Computing Essentials, http://www.mupad.com/majewski/edition1/index.html, 2004.
7. Z. Nowakowski, Dydaktyka informatyki w praktyce (Polish) [Teaching Informatics in Practise], MIKOM, Warszawa 1996.
8. W.Oevel, S.Wehmeier, J. Gerhard, The MuPAD Tutorial, MuPAD(TM) Light Version 2.5.3.

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Author's address

Danuta Rozpłoch - Nowakowska
Faculty of Mathematics and Computer Science
Nicolaus Copernicus University
ul. Chopina 12/18, 87-100 Toruń
Poland
e-mail: nowa at mat.uni.torun.pl

All interactive images in this article were produced with MuPAD and JavaView by M.Majewski