Triple integrals

If $f(x,y,z)$ is continuous on a box region $E=[a,b] \times[c,d] \times [e,f]$, then

$\int \int \int_E f(x,y,z) dV = \int^f_e \int^d_c \int^b_a f(x,y,z)dxdydz$

Suppose we are integrating $f(x,y,z)$ on region $E$. There are 3 cases of finding region $E$.

Case 1: Region $E$ is

$(x,y) \in D$
$h_1(x,y) \leq z \leq h_2(x,y)$

So,

$V = \int \int \int_E f(x,y,z)dV = \int \int_D [ \int^{h_2(x,y)}_{h_1(x,y)} f(x,y,z) dz]dA$

Case 2: Region $E$ is

$(y,z) \in D$
$h_1 (y,z) \leq x \leq h_2(y,z)$

So,

$V = \int \int \int_E f(x,y,z)dV = \int \int_D [ \int^{h_2(y,z)}_{h_1(y,z)} f(x,y,z) dx]dA$

Case 3: Region $E$ is

$(x,z) \in D$
$h_1 (x,z) \leq y \leq h_2 (x,z)$

So,

$V = \int \int \int_E f(x,y,z)dV = \int \int_D [ \int^{h_2(x,z)}_{h_1(x,z)} f(x,y,z) dy]dA$

The volume of a 3D region $E$ is given by the triple integral:

$V = \int \int \int_E dV$