Printing that is the easy part:

What you want to do with the result is something else.

@erik10 See if this is what you want.

Note: The message on the output says "number of chars = 22".  The "22" is shown properly in the worksheet but comes out broken on this website. Not my fault.

Download mw.mw

Download mw.mw

If you solve the equation x^2 + y^2 = 4 for y, you will get two solutions, y=±sqrt(4 - x^2). The plus sign gives the equation of the upper half of the circle, and the minus sign gives the equation of the lower half. So the answer to Part (a) is y = sqrt(4 - x^2), and the plot is produced by

plot(sqrt(4 - x^2), x=-2..2, scaling=constrained);

As to part (b): Rotating that semicircle about the x axis produces a sphere of radius 2.

As to part (c): The volume enclosed by the sphere is given by the formula V_x that you have shown in your post. We have f(x) = sqrt(4 - x^2), and therefore f(x)^2 = 4 - x^2.  So the volume is:

V__x := Pi*int( 4 - x^2, x = -2 .. 2);

I haven't examined / debugged your code since it seems to me much too complicated for what it's supposd to do. Instead, I wrote the following alternative which does what you want. This is a slightly edited version of what I had initially posted.

Download piston-crank-animation2.mw

The format of the font specification is defined in ?plot,options under the "font" entry. Your specification axesfont=[12,12] does not fit that specification. Maple does not complain about that in the worksheet mode. I would call that a shortcoming of the worksheet-mode parser.  The commandline-mode parser correctly flags it as invalid.

I suppose that by axesfont=[12,12]  you wish to select 12-point fonts for the horizontal and vertical axes.  To see that Maple's worksheet-mode parser interprets that differently, change that to axesfont=[36,8].  You will see that the "36" is ignored.

The try/catch mechanism may be what you are looking for.

Download mw.mw

Your code expresses the essence of the solution, but it can be streamlined a bit.

restart;
eq := <x, y> =~ lambda*<1, 1> + mu*<1, 2>;
convert(eq, set);
solve(%, {lambda, mu});

We see that solve() returns a unique solution, indicating that any vector <x,y> may be uniquely represented as a linear combination of <1,1> and <1,2>, indicating that those form a basis for the space.

Maple's PDF export has been broken in the last several releases. Despite repeated complaints on this forum, no solution seems to be forthcoming.

These two workarounds work quite well on the command-line in Linux.  The equivalent command lines may be available in Mac and Windows but I have neither so I wouldn't know.

1. Workaround #1:  Save the graphics as EPS, let's say filename.eps, then do
       epstopdf filename.eps
   to produce filename.pdf.
2. Workaround #2:  Save the graphics as SVG, let's say filename.svg, then do
       inkscape --export-type=pdf --export-area-drawing filename.svg
   to produce filename.pdf.

Both methods produce good quality PDF.  The file size of the PDF file in method #2 is about 1/20 of that of method #1 but the latter is more faithful to the original.

I have attached PDFs cooresponding to 
    plot3d(x^2+y^2, x=-1..1, y=-1..1);
converted through the two methods.

parabola-from-eps.pdf
parabola-from-svg.pdf

Here is the solution, but don't just pass it on to your teacher. She won't believe you.

To get an answer with detailed explanation, show what you know and how far you have gotten on your own on this homework.

restart;
f := x -> (x^3 + 9*x^2 - 9*x - 1) / (x^4 + 1);
numer(D(f)(x) - 1/2);
xvals := fsolve(%, x, maxsols=4);
seq(f(t) + 0.5*(x-t), t in xvals);
plot([f(x), %], x=-7..7, scaling=constrained, color=[seq(cat("oldplots ", i), i=1..5)]);

restart;
with(plots):
with(plottools):
p1 := plot(sin(x), x=-2*Pi..2*Pi, color="Red"):
display(
	plot(sin(x), x=-2*Pi..2*Pi, color="Green"),
	line([-2*Pi,0], [2*Pi,0]),
	line([0,-2*Pi], [0,2*Pi])
):
p2 := rotate(%, Pi/4):
display(p1,p2, scaling=constrained);

I don't see a simple way of adding tickmarks.

restart;
with(plots):
vertices := [
	c__1 = <0, -2>,
	c__2 = <1, 2>,
	c__3 = <2, 2>,
	c__4 = <0.5, 6>,
	c__5 = <1, 4> 
]:
Labels := map( v -> [convert(rhs(v), list)[], lhs(v)], vertices):
display(
	textplot(Labels, align={above}),
	plottools:-polygon(convert~(rhs~(vertices), list), style=line, color=red)
);

See https://www.mapleprimes.com/questions/229945-Simulation-Of-A-Solution-Of--Neutral for a lot of ideas on how to handle delay differential equations.

I don't see what you mean by tracing the moving points since the number of point changes in your sequence of plots. Here is how to draw traces of individual moving points that preserve their identities during the motion.

restart;
with(plots):
f := t -> sin(t);
g := t -> [cos(t), sin(t)];
trace_me := proc(t)
	display(
		pointplot([t,f(t)], symbol=solidcircle, symbolsize=30, color="Red"),
		plot(f(s), s=0..t, color="Green"),
		pointplot(g(t), symbol=solidcircle, symbolsize=30, color="Orange"),
		plot([g(s)[], s=0..t], color="Blue"),
	scaling=constrained);
end proc:
animate(trace_me, [t], t=0..2*Pi);