@JAMET Put your Replies after the Answer that they pertain to.

@JAMET The code above will only run in 1D input.

@dingtianlidi It's true that some first-year calculus textbooks give an simplified definition of limit that requires that the function be defined in some open interval containing the limiting value of the independent variable (except possibly at that limiting value itself). If this is the definition that you follow, then it's true that the limit doesn't exist. However, this is not the traditional definition, which allows the domain D of the function to be completely arbitrary.

@dingtianlidi The precise traditional epsilon-delta definition of limit is given on this Wikipedia page. See the section "Precise statement for real-valued functions". Notice that it allows the domain D of the function to be completely arbitrary. In particular the function may be undefined at an infinite number of points in every neighborhood of the limiting value of the independent variable.

@anthonyfl Regardless of whether you explicitly use coords= cylindrical (as I did), or explicitly use the coordinate transformations {z= r*cos(theta), x= r*sin(theta), y=y} as Kitonum and Acer did, and regardless of whether you want shells or washers, doing these animations requires that you "think" in cylindrical coordinates. Doing the integrals, however, doesn't require sophisticated 3-D thinking.

There is no attached file to your Question. Please try attaching again.

This comment is unrelated to your Question: Why do you want to do a log plot (as opposed to a regular plot) over such a small range? The total variation of the t-axis is only 1.52/1.4 = 9%.

@Anthrazit The command trunc is better than round for this purpose because it's built-in. 

How many nodes have degree 2?

Are you trying to plot it, animate it, or find the volume?

@Kitonum To complete Kitonum's list, we must represent 188 as a sum of 6 distinct squares: 188 = 1 + 4 + 9 + 25 + 49 + 100. This is the only such representation. There is no representation using fewer distinct squares.

@anthonyfl It's just an isometry, so relabel the axes:

plots:-animate(
    plots:-display, 
    ['''plot3d''']~(
        [[r,t,r^2/8], [r,t,sqrt(r)]], 
        r= 0..4, t= 0..T, coords= cylindrical, 
        labels= ['x', 'z', 'y'], axes= normal, scaling= constrained,
        view= [-4..4, -4..4, 0..4]
    ),   
    T= 0..2*Pi, frames= 60, paraminfo= false
);

@anthonyfl It might be due to having an older version of Maple. Try this:

plots:-animate(
    plots:-display, 
    [''plot3d''~(
        [[r,t,r^2/8],[r,t,sqrt(r)]], 
        r= 0..4, t= 0..T, coords= cylindrical
    )],   
    T= 0..2*Pi, frames= 60
);

I think that that'll work in any version since Maple 13. Note that the quotes on plot3d are two pairs of single quotes, not one pair of double quotes.

@Carl Love There's another special case that's worth mentioning, especially because it comes up fairly often: If the value assigned to a name is a procedure (including arrow expressions such as x-> x^2), a module (including Records) that's not an object, or a table, then the default assignment is by address (perhaps the correct term is "by reference", as Acer says, but I prefer "address"). If you want the value in those cases. then you need eval. Examples:

restart:
A:= x-> 7*x:  B:= A:  A:= x-> 9*x:  C:= A:  B(x), C(x);
                            9 x, 9 x
restart:
A:= x-> 7*x:  B:= eval(A):  A:= x-> 9*x:  C:= A:  B(x), C(x);
                            7 x, 9 x

See help page ?last_name.

Regarding initialization of names: Any name's initial value is just the name itself, or, if you prefer, its address. The use of names with no other value than themselves is fundamental to Maple programming. 

@John2020 You're welcome. You could show your appreciation by giving the Answer a Vote Up by clicking on the thumb icon in the upper right corner.