# Triple integrals

### Triple integrals

#### Lessons

Triple Integrals with a Box Region

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

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

3 Cases of General Regions

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 Use of Triple Integrals

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

$V = \int \int \int_E dV$

• Introduction
Triple Integrals Overview:
a)
• Similar to rectangular regions, but 3D
• $[a,b]$ is the $x$ part
• $[c,d]$ is the $y$ part
• $[e,f]$ is the $z$ part
• Integral signs are interchangeable
• An Example

b)
3 Cases of General Regions
• Case 1: $(x,y) \in D,h_1(x,y) \leq z \leq h_2(x,y)$
• Case 2: $(y,z)\in D, h_1(y,z) \leq x \leq h_2(y,z)$
• Case 3: $(x,z)\in D, h_1(x,z) \leq y \leq h_2(x,z)$
• An Example

c)
Use of Triple Integrals
• Finds volume in 3D space
• An Example

• 1.
Evaluate

$\int_{0}^{2}\int_{1}^{2}\int_{-1}^{1} 2xy^{2} - zdzdydx$

• 2.
Evaluate

$\int_{0}^{\pi}\int_{0}^{z}\int_{0}^{x} sin \, x dydxdz$

• 3.
Finding the Volume of 3D objects
Use triple integrals to determine the volume of the region below $\, z = 2 \,$, above $\, z = 1\,$, bounded by $\, y = x^{2} - 1 \,$ and $\, y = 1 - x^{2}$

• 4.
Evaluating Triple Integrals with Different types of Region E
Evaluate $\, \int\int\int_{E} \, xd V \,$ where $\, E \,$ is the region bounded by $\, z = x^{2} + y^{2} -2 \,$ and the plane $\, z = 2$.

• 5.
Evaluate $\, \int\int\int_{E} \, 2 + ydV\,$ where $\, E \,$ is the region below $\, xy + 3 \,$ above the region $\, z = 2$, and bounded by $\, 0 \leq x \leq 1, 0 \leq y \leq 1$