3. Triple Integrals Over General Regions

Referring to Type I and Type II regions from Double Integrals ⇒ 3 Double Integrals Over General Regions

Regions of Type I

Definition

A solid region $E$ is said to be of type 1 or $z$-simple if it lies between the graphs of two continuous functions of $x$ and $y$:

$E=\{(x,y,z)|(x,y)\in D,u_1(x,y)\le z\le u_2(x,y)\}$

where $D$ is the projection of $E$ onto the $xy$-plane.

Triple Integrals Over a Region of Type I

Theorem

If the solid region $E$ is of type 1, i.e.

$E=\{(x,y,z)|(x,y)\in D,u_1(x,y)\le z\le u_2(x,y)\},$

then

${\iiint }_{E}f(x,y,z)dV={\iint }_D\left({\int }_{u_1(x,y)}^{u_2(x,y)}f(x,y,z)dz\right)dA$

Regions of Type II

Definition

A solid region $E$ is said to be of type 2 or $x$-simple if it lies between the graphs of two continuous functions of $y$ and $z$:

$E=\{(x,y,z)|(y,z)\in D,u_1(y,z)\le x\le u_2(y,z)\},$

where $D$ is the project of $E$ onto the $yz$-plane.

Triple Integrals Over a Region of Type II

Theorem

If the solid region $E$ is of type 2, i.e.

$E=\{(x,y,z)|(y,z)\in D,u_1(y,z)\le x\le u_2(y,z)\},$

then

${\iiint }_{E}f(x,y,z)dV={\iint }_D\left({\int }_{u_1(y,z)}^{u_2(y,z)}f(x,y,z)dx\right)dA$

Regions of Type III

Definition

A solid region $E$ is said to be of type 3 or $y$-simple if it lies between the graphs of two continuous functions of $x$ and $z$:

$E=\{(x,y,z)|(x,z)\in D,u_1(x,z)\le x\le u_2(x,z)\},$

where $D$ is the project of $E$ onto the $xz$-plane.

Triple Integrals Over a Region of Type II

Theorem

If the solid region $E$ is of type 3, i.e.

$E=\{(x,y,z)|(x,z)\in D,u_1(x,z)\le y\le u_2(x,z)\},$

then

${\iiint }_{E}f(x,y,z)dV={\iint }_D\left({\int }_{u_1(x,z)}^{u_2(x,z)}f(x,y,z)dy\right)dA$