# Thread: need help on this integral~

1. ## need help on this integral~

$\displaystyle \int \int \int xy^2z^3 \,dx \,dy \,dz$ , where D is the solid in R3 bounded by the surface z = xy
and the planes x = y, x = 1, and z = 0.

I couldn't figure out the right limit.

but I tried, the inner one is from z=0 to z=xy

the middle one is from y=1 to y

the outter one is from 0 to 1

am i right?
if not , can anyone explain to me?

2. Originally Posted by yzc717
$\displaystyle \int \int \int xy^2z^3 \,dx \,dy \,dz$ , where D is the solid in R3 bounded by the surface z = xy
and the planes x = y, x = 1, and z = 0.

I couldn't figure out the right limit.

but I tried, the inner one is from z=0 to z=xy (*)

the middle one is from y=1 to y (*)

the outter one is from 0 to 1 (*)

am i right?
if not , can anyone explain to me?
You have to be careful on the order of $\displaystyle dx\, dy\, dz$ in your integral, typically we have

$\displaystyle \int_{P1}^{P_2} \int_{C_1}^{C_2} \int_{S_1}^{S_2} f(x,y,z)\,dV$

where $\displaystyle S_1 \to S_2$ surface to surface,
where $\displaystyle C_1 \to C_2$ curve to curve and
where $\displaystyle P_1 \to P_2$ point to point.

The order of $\displaystyle dx\,dy\,dz$ in $\displaystyle dV$ gives the direction. So in

$\displaystyle \int \int \int xy^2z^3 \,dx \,dy \,dz$ the dx first is giving the surface to surface in the x direction. However, when giving your limits (in blue above) is suggests that what you're doing is the following

$\displaystyle \int_{x=0}^{x=1} \int_{y=0}^{y=x} \int_{z=0}^{z=xy} xy^2z^3 \,dz \,dy \,dx$

but we typically write

$\displaystyle \int_0^1 \int_0^x \int_0^{xy} xy^2z^3 \,dz \,dy \,dx$

3. Thank you , its now really make sense