I have bee given the problem:

A flying saucer has shape bounded below and above by the functions:

$\qquad f(x,y) = 61/5 - 12x/5 + x^2 - 12y/5 +6xy/5 +2y^2/5$


$\qquad g(x,y) = 15 - x^2 - 6xy/5 - 2y^2/5$

If the density of the saucer is $(x,y,z) = x^2 + 3z$, find the mass.

The example we were given related a different function to a numerical value, 10, and used Solve[... == 10, y] to find solutions to the quadratic in terms of y as a function of x, but this isn't applicable to the situation I have, so I'm kind of stuck.


2 Answers 2


In Mathematica I would use Integrate with ImplicitRegion. First, the ImplicitRegion:

region = ImplicitRegion[
        61/5 - 12 x/5 + x^2 - 12 y/5 + 6 x y/5 + 2 y^2/5,
        15 - x^2 - 6 x y/5 - 2 y^2/5
Region @ region

enter image description here

Now, for the integral:

    x^2 + 3z,
    {x, y, z} ∈ region

34048 π/15


It sounds like you intend to use Solve to find the limits of integration over the density. If you define f and g in Mathematica:

f[x_,y_]:=61/5 - 12 x/5+x^2-12 y/5+6 x y/5+2 y^2/5;
g[x_,y_]:=15-x^2-6 x y/5-2 y^2/5;

And solve them for each other:


You'll find the limits for y in terms of x. In the integral, you'll use it as {y,(3-3x-Sqrt[23-6x-x^2])/2,(3-3x+Sqrt[23-6x-x^2])/2}, taking note that the subtracted square root is always going to be the lower bound.

Use the determinant of y's solution to find the boundaries on x:


The limits on x are {x,-3-4Sqrt[2],-3+4Sqrt[2]}. Then use Integrate on the density over the enclosed volume, making sure to put the x term before the y term, since the y term depends on x:


Which should be 34048 Pi/15.

Mathematica can figure out this condition on its own as well:

Integrate[(x^2+3z) Boole[f[x,y]<z<g[x,y]],{x,-Infinity,Infinity},{y,-Infinity,Infinity},{z,-Infinity,Infinity}]

The Boole term is 1 when the condition f[x,y]<z<g[x,y] is true and 0 otherwise, so the integral can be performed over all of space. This should get the same answer.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.