- #1
jonroberts74
- 189
- 0
Homework Statement
I am given
[tex]W = \{ (x,z,z)| \frac{1}{2} \le z \le 1; x^2 + y^2 +z^2 \le 1\}[/tex]
they want the iterated integrals to be of the form
[tex]\iiint_W dzdydx[/tex]
The Attempt at a Solution
so I know z=1/2 will give me the larger bound for x
[tex]x^2 + y^2 + (1/2)^2 =1 \rightarrow x^2 + y^2 = 3/4[/tex]
[tex]x^2 + y^2 = 3/4[/tex] gives me [tex]- \sqrt{3/4} \le x \le \sqrt{3/4}[/tex] when y=0
so y bounds are
[tex]- \sqrt{3/4-x^2} \le y \le \sqrt{3/4-x^2}[/tex]
and my z bounds
[tex]\sqrt{1-y^2} + \frac{1}{2} \le z \le \sqrt{1-y^2} [/tex]
so I get
[tex]\int_{- \sqrt{3/4}}^{\sqrt{3/4}}\int_{- \sqrt{3/4-x^2}}^{\sqrt{3/4-x^2}}\int_{\sqrt{1-y^2} + \frac{1}{2}}^{\sqrt{1-y^2}}f(x,y,z)dzdydx[/tex]