# triple integral in given region

• November 16th 2009, 08:03 PM
zpwnchen
triple integral in given region
what are the limit of z,y and x to integrate?

I got
$-sqrt(1-y^2) \leq z \leq sqrt(1-y^2)$
$-1 \leq y \leq 1$
$0 \leq x \leq 9$

help me
• November 17th 2009, 12:33 AM
simplependulum
hi

I perfer to transform the rectangular coordinates to polar ( cylindrial )

but before changing $dxdydz$ into $rdrd{\theta}dz$

we first exchange $x$ and $z$, it would be easier for us to solve .

Therefore , the boundary becomes $z = 9r^2 , z = 9$

and when $z = 9$ , $9 = 9 r^2 \implies r=1$

the integral becomes $\int\int\int_{E_1} z rdrd{\theta}dz$

$= \int_0^{2\pi} \int_0^1 \int_{9r^2}^{9} zr dzdrd{\theta}$

$= 2\pi \int_0^1 r \frac{1}{2} (9^2 - 9^2 r^4 ) ~dr$

$= 81\pi \int_0^1 (r - r^5)dr = 27 \pi$
• November 17th 2009, 12:39 AM
simplependulum
Quote:

Originally Posted by zpwnchen
what are the limit of z,y and x to integrate?

I got
$-sqrt(1-y^2) \leq z \leq sqrt(1-y^2)$
$-1 \leq y \leq 1$
$0 \leq x \leq 9$

help me

It should be

$\int_{-1}^1 \int_{-\sqrt{1-z^2}}^{\sqrt{1-z^2}} \int_{9y^2 + 9z^2}^{9} x dxdydz$