Q:

Sketch f(x,y)=(21/4)x^2y over the region x^2 <= y <= 1

Can someone share a technique using MATLAB to plot the surface f(x,y)=(21/4)x^2y over the region x^2 <=  y <= 1?
Also, if anyone is aware of some tutorials or links that would help with this type of problem, could you please share them?
Thanks.
Here is another approach:
%%
close all
x=linspace(-1,1,40);
g1=x.^2;
g2=ones(1,40);
y=[];
n=20;
for k=0:n
    y=[y;g1+(g2-g1)*k/n];
end
x=x(ones(1,n+1),:);
z=21/4*x.^2.*y;
meshz(x,y,z)
axis tight
xlabel('x-axis')
ylabel('y-axis')
view(136,42)

And the result:

And finally, you can map the region (-1,1)x(0,1) in the uv-plane into the region bounded by $y=x^2 and y=1 in the xy-plane with the parametrization:
f(u,v) = (u\sqrt{v},v)
Capture from: https://math.stackexchange.com/questions/823168/transform-rectangular-region-to-region-bounded-by-y-1-and-y-x2
This code produces the same image shown above:
close all
[u,v]=meshgrid(linspace(-1,1,40),linspace(0,1,20));
x=u.*sqrt(v);
y=v;
z=21/4*x.^2.*y;
meshz(x,y,z)
axis tight
xlabel('x-axis')
ylabel('y-axis')
view(136,42)

A:

First off, let's look at your valid region of values.  This is telling us that y >= x^2 and also y <= 1.  This means that your y values need to be on the positive plane bounded by the parabola x^2 and they also must be less than or equal to 1.  In other words, your y values must be bound within the area dictated from y = x^2 to y = 1.  Pictorially, your y values are bounded within this shape:

As such, your x values must also be bound between -1 and 1.  Therefore, your actual boundaries are: -1 <= x <= 1 and 0 <= y <= 1.  However, this only locates our boundaries for x and y but it doesn't handle where the plot has valid values.  We'll tackle that later.  
Now that we have that established, you can use ezsurf to plot surface plots in MATLAB that are dictated by a 2D equation.
You call ezsurf like so:
ezsurf(FUN, [XMIN,XMAX,YMIN,YMAX]);

FUN is a function or a string that contains the equation you want, and XMIN,XMAX,YMIN,YMAX contain the lowest and highest x and y values you want to plot.  Plotting without these values assumes a span from -2*pi to 2*pi in both dimensions.  As such, let's create a new function that will handle when we have valid values, and when we don't.  Use this code, and save it to a new file called myfun.m.  Make sure you save this to your current Working Directory.
function z = myfun(x,y)
    z = (21/4)*x.^2.*y;
    z(~(x.^2 <= y & y <= 1)) = nan;
end

This will allow you to take a series of x and y values and output values that are dictated by the 2D equation that you have given us.  Any values that don't satisfy the condition of x^2 <= y <= 1, you set them to NaN.  ezsurf will not plot NaN values.
Now, call ezsurf like so:
ezsurf(@myfun, [-1,1,0,1]); 

You thus get:

This will spawn a new figure for you, and there are some tools at the top that will allow you interact with your 3D plot.  For instance, you can use the rotation tool that's at the top bar beside the hand to rotate your figure around and see what this looks like.  Click on this tool, then left click your mouse and hold the left mouse button anywhere within the surface plot.  You can drag around, changing the azimuth and the latitude to get the perspective that you want.
Edit: June 4th, 2014
Noting your comments, we can decrease the jagged edges by increasing the number of points in the plot.  As such, you can append a final parameter to ezsurf which is N, the number of points to add in each dimension.  Increasing the number of points will decrease the width in between each point and so the plot will look smoother.  The default value of N is 60 in both dimensions.  Let's try increasing the amount of points in each dimension to 100.
ezsurf(@myfun, [-1,1,0,1], 100);

Your plot will look like:

Hope this helps!

