Multivariable Calculus Problem, Struggling with final question

[doongle12345]

Homework Statement

The body B = f(x; y; z) belongs to R3,
x^2+4y^2+3z^2<or equal to 3,
x> or equal to 0
y> or equal to 0
z> or equal to 0
The body B has a volume density:
M(x,y,z)= (xyz^2)/(1+x^2+4y^2+3z^2)

Calculate the total mass and average density of the body.

Sorry if non-standard notation has been used, was difficult to write, struggling with this problem, thanks for any help :) x

 

[bossman27]

Here's a couple of hints:

Mass = \int \int \int_{B} M(x,y,z) dx dy dz

Your limits of integration will be of the form:

u_{1}(y,z) \leq x \leq u_{2}(y,z)

h_{1}(z) \leq y \leq h_{2}(z)

c \leq z \leq d where d, c are constants.

Think of these functions as the "boundaries" that enclose the given variable. To start you off, it is easy to show that:

z_{max}, z_{min} will occur when x, y = 0

From our equation describing B, plugging in x=y=0:
z_{max} = 1; z_{min} = -1

See if you can figure out the limits of integration for y and x.

Note: you could also do it in the order dz dx dy, dz dy dx, etc. It does't really matter, although the constants for your third integration will be simplest in this case if you use z last.