@Markiyan Hirnyk 

I'm split book from page 187 to 217:Pages-187-217.pdf

@Markiyan Hirnyk 

You can download book from : http://www.if.ufrj.br/~tgrappoport/aulas/metfis2/0123747325.pdf

@_Maxim_ 

Yes You a right.   

evala(Normal(simplify(diff(int(sqrt(t^4+1)-t^2, t), t))))
# sqrt(t^4+1)-t^2

This is incorrect:

evalf(int(sqrt(t^4+1)-t^2, t = 1 .. infinity));
# -1.992145865+3.*10^(-10)*I

It should be :

#0.4799537050

Yes is a bug in int function.

expand(simplify(diff(int(sqrt(t^4+1)-t^2, t), t)));

# t^4/sqrt(t^4+1)-t^2+1/sqrt(t^4+1)

I will submit this bug to SCR.

I think is not a bug.

Try:

pdsolve([diff(u(t, x), t, t) = diff(u(t, x), x, x), u(t, 0) = 0, u(t, Pi) = 0, u(0, x) = 0]);

#u(t, x) = Sum(_C1[n]*sin(n*t)*sin(n*x), n = 1 .. infinity)

C1[n] and C2[n]  are dependent on boundaryconditions for time variable (u(T1,x)=u1,u(T2,x)=u2).

maybe I'm wrong....

@markweitzman

Eigen4 := (dsolve({bc, ode}, numeric, range = 0 .. 2, maxmesh = 8192, abserr = 1.*10^(-1), approxsoln = [y(u) = -u, e = 2]))(0)[4];
Eigen5 := (dsolve({bc, ode}, numeric, range = 0 .. 2, maxmesh = 2192, abserr = 1.*10^(-3), approxsoln = [y(u) = sin(u), e = 2]))(0)[4];

You have three conditions and two degrees of freedom. There may not be a solution.

Use:

pdsolve([Yours System-PDE]);

@vv

For Maple users it will be useful to someone and for me.

Thanks a lot :)

Mathematica also no better.

@Kitonum 

Works Fine on my system:

`Standard Worksheet Interface, Maple 2017.3, Windows 8.1, September 27 2017 Build ID 1265877`

`Maple 2017.3, X86 64 WINDOWS, Sep 27 2017, Build ID 1265877`

NO_ERROR.mw

Thanks to share a cool stuff.

Maybe you can add(improve) to calculate surface area if  it's possible.

SurfaceArea(1, [x^2+y^2+z^2 <= 1, 5*x^2-z^2 <= 1], [x, y, z])

# 6.50063

Mathematica solution:

@vv 

Yes you are right,I made a mistake.Now its work fine.

Thanks you for your feedback.

@vv 

Good Job and great code,but only work for simple cases.

R := {(16*x^5-20*x^3+5*x)^2-(16*x^5-20*x^3+5*x)*(8*y^4-8*y^2+1)+(8*y^4-8*y^2+1)^2 <= -3/4}
Area(R)

With this code Maple give a 0 and Mathematica an error.

This is a very good question.

I putt in Google "region measure in MAPLE" and  I'm  not find anything conclusive.

In Maple  there no such built functions like in Mathematica to measure region.

Maybe in next  release in Maple 2017.4  we will see these function. Maybe !