Section 4.4 Triple Integrals

4.4 Triple Integrals 

In the real world, the integration can be used not only the computation of volumes but also computation of total heat or quantity over a space.

It makes sense to ask that can we do integration over a solid, i.e. triple integral. The idea is exactly the same as double integrals or single integrals, Riemann sum. 

Definition: Triple Integrals

The triple integral of a function $f(x,y,z)$ over the a rectangular box $B$ is defined as

$$\int \int {\int }_{B}f(x,y,z)dV=\underset{l,m,n\to \mathrm{\infty }}{lim}\sum _{i=1}^l\sum _{j=1}^m\sum _{k=1}^{n}f(x_{ijk}^{\ast },y_{ijk}^{\ast },z_{ijk}^{\ast })\mathrm{△}V$$

if the limit exists.

Theorem: Fubini’s Theorem for Triple Integrals 

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

$$\int \int {\int }_{B}f(x,y,z)dV={\int }_e^f{\int }_c^d{\int }_a^{b}f(x,y,z)dxdydz.$$

This integral is also equal to any of the other five possible orderings for the iterated triple integral.

Example 1: Evaluating the triple integral ${\int }_0^1{\int }_2^4{\int }_{-1}^3(x+yz)dxdydz$.

Exercise 1: Evaluating the triple integral ${\int }_{-1}^1{\int }_0^2{\int }_1^4(xy-z)dxdydz$.

Remark (Theorem): As before, we can do the integration over a general box, not just a rectangular one. Let $E=\{(x,y,z)|a\le x\le b,g_1(x)\le y\le g_2(x),u_1(x,y)\le z\le u_2(x,y)\}$ and $f(x,y,z)$ be a continuous function over $E$, then 

$$\int \int {\int }_{E}f(x,y,z)dV={\int }_a^b{\int }_{g_1(x)}^{g_2(x)}{\int }_{u_1(x,y)}^{u_2(x,y)}f(x,y,z)dzdydx$$

Example 2: Evaluating the triple integral ${\int }_0^1{\int }_z^{2z}{\int }_0^{y+z}2ydxdydz$.

Exercise 2: Evaluating the triple integral ${\int }_0^1{\int }_y^{3y}{\int }_0^{x-y}xdzdxdy$.

Example 3: Evaluating the triple integral of the function $f(x,y,z)=2x-y$ over the solid bounded by the planes $x=0$, $y=0$, $z=0$ and $x+y+z=1$. 

Exercise 3: Evaluating the triple integral of the function $f(x,y,z)=y+z$ over the solid bounded by the planes $x=0$, $y=0$, $z=0$ and $x-y-z=1$. 

Example 4: Evaluating the triple integral of the function $\int \int {\int }_E(x+y)dV$ where $E$ is the solid bounded by $z=y^2-1$, $z=7-y^2$, $x=0$ and $x=2$. 

Exercise 4: Evaluating the triple integral of the function $\int \int {\int }_E(z-y)dV$ where $E$ is the solid bounded by $x=y^2-5$, $x=13-y^2$, $z=0$ and $z=1$. 

Example 5: Evaluating the triple integral of the function $\int \int {\int }_E\sqrt{x^2+z^2}dV$ where $E$ is the solid bounded by $y=x^2+z^2$, and $y=1$.

Exercise 5: Evaluating the triple integral of the function $\int \int {\int }_E\sqrt{y^2+z^2}dV$ where $E$ is the solid bounded by $x=y^2+z^2$, and $x=4$.

Example 6: Find the volume of the solid bounded by $x=y^2+z^2$ and $x=32-y^2-z^2$.

Exercise 6: Find the volume of the solid bounded by $z=y^2+x^2$ and $z=50-y^2-x^2$.

Example 7: Find the volume of the solid bounded by $y=x^2$, $z=0$ and $y+z=1$.

Example 8: Evaluating the triple integral of the function $\int \int {\int }_E{y}^{2}dV$ where $E$ is the solid tetrahedron with vertices $(0,0,0)$, $(2,0,0)$, $(0,2,0)$ and $(0,0,2)$. 

Group work:

1. Find the volume of the solid bounded by $z=y^2$, $x=0$ and $z+x=3$.

2. Find the volume of the solid bounded by $x^2+y^2=1$, $z=0$ and $y+z=3$.

3. Evaluating the triple integral of the function $\int \int {\int }_{E}xdV$ where $E$ is the solid below the plane $z=y$ and above the triangular region with vertices $(0,0,0)$, $(1,0,0)$ and $(0,1,0)$.