Is My Triple Integral Calculation Correct or Is There an Error in the Book?

[mikan]

Hi,
my result of

$$\int \int \int_{A} xyz dxdydz$$

where

$$A = \{(x,y,z); x^2+y^2+z^2 \leq 2, x \geq 0, y \geq 0, z \geq 0 \}$$

is

$$\frac {8}{48}$$,

but book says

$$\frac{1}{48}$$.

Is the book right? Could you please verify?

Thank you
Michael

 

[driscoll79]

Hey Michael,

I also got 8/48 using:

$$x = \rho \sin\phi \cos\theta, y = \rho \sin\phi \sin\theta, z = \rho \cos\phi$$ with $$0 < \phi < \frac{\pi}{2}, 0 < \theta < \frac{\pi}{2}, 0 < \rho < \sqrt{2}$$

so it looks to be correct, although I'd suggest checking everything one more time just to be sure.