1) Introduction: Entering Three-Dimensional Integration 📚
After mastering double integrals, we now extend integration to three dimensions with triple integrals. These calculate volumes, masses, and other properties over 3D regions, such as solids in space. Triple integrals are vital in physics (e.g., gravitational fields), engineering (e.g., material density), and geometry. This topic will guide you from the basics to advanced techniques, ensuring a complete understanding.
We’ll cover:
- Definition and Notation: Understanding triple integrals and their setup.
- Computation Over Rectangular Boxes: Integrating over simple 3D regions.
- Non-Rectangular Regions: Handling complex solids.
- Applications: Volume, mass, and center of mass in 3D.
Quick Recap: A double integral \( \iint_R f(x, y) \, dA \) extends to a triple integral \( \iiint_E f(x, y, z) \, dV \) for 3D volumes.
2) Definition and Notation 🎓
A triple integral \( \iiint_E f(x, y, z) \, dV \) computes the signed volume under a 3D surface or a property (e.g., mass) over a region \( E \) in space. It’s the limit of Riemann sums over a 3D grid.
Definition 21.1: Triple Integral
For a function \( f(x, y, z) \) over a region \( E \), the triple integral is \( \iiint_E f(x, y, z) \, dV = \lim_{m, n, p \to \infty} \sum_{i=1}^m \sum_{j=1}^n \sum_{k=1}^p f(x_{ijk}, y_{ijk}, z_{ijk}) \Delta V \), where \( \Delta V \) is the volume of each subbox.
Example: \( \iiint_E 1 \, dV \) over \( E = [0, 1] \times [0, 1] \times [0, 1] \).
**Iterated Integrals**: Computed as \( \int_a^b \int_c^d \int_e^f f(x, y, z) \, dz \, dy \, dx \), with order flexibility if \( f \) is continuous.
Example 1: Triple Integral Over a Box
Evaluate \( \iiint_E 1 \, dV \) where \( E = [0, 2] \times [0, 1] \times [0, 3] \).
- This is the volume: \( \int_0^2 \int_0^1 \int_0^3 1 \, dz \, dy \, dx \).
- Innermost: \( \int_0^3 1 \, dz = 3 \).
- Middle: \( \int_0^1 3 \, dy = 3 \).
- Outer: \( \int_0^2 3 \, dx = 6 \).
Answer: 6 (volume of box).
Example 2: Constant Function
Compute \( \iiint_E 2 \, dV \) over \( E = [-1, 1] \times [0, 2] \times [0, 1] \).
- \( \int_{-1}^1 \int_0^2 \int_0^1 2 \, dz \, dy \, dx \).
- Innermost: \( \int_0^1 2 \, dz = 2 \).
- Middle: \( \int_0^2 2 \, dy = 4 \).
- Outer: \( \int_{-1}^1 4 \, dx = 8 \).
Answer: 8.
3) Computation Over Rectangular Boxes 📐
For a rectangular box \( E = [a, b] \times [c, d] \times [e, f] \), the triple integral is an iterated integral. The order of integration (dx dy dz, dy dz dx, etc.) can be adjusted if \( f \) is continuous.
Definition 21.2: Iterated Triple Integral
\( \iiint_E f(x, y, z) \, dV = \int_a^b \int_c^d \int_e^f f(x, y, z) \, dz \, dy \, dx \), with possible order changes.
Example 3: Linear Function
Evaluate \( \iiint_E (x + y + z) \, dV \) over \( E = [0, 1] \times [0, 1] \times [0, 1] \).
- \( \int_0^1 \int_0^1 \int_0^1 (x + y + z) \, dz \, dy \, dx \).
- Innermost: \( \int_0^1 (x + y + z) \, dz = [xz + yz + \frac{z^2}{2}]_0^1 = x + y + \frac{1}{2} \).
- Middle: \( \int_0^1 (x + y + \frac{1}{2}) \, dy = [xy + \frac{y^2}{2} + \frac{y}{2}]_0^1 = x + \frac{1}{2} + \frac{1}{2} = x + 1 \).
- Outer: \( \int_0^1 (x + 1) \, dx = [\frac{x^2}{2} + x]_0^1 = \frac{1}{2} + 1 = \frac{3}{2} \).
Answer: \( \frac{3}{2} \).
Example 4: Changing Order
Compute \( \iiint_E xyz \, dV \) over \( E = [0, 1] \times [0, 2] \times [0, 1] \) both ways.
- \( \int_0^1 \int_0^2 \int_0^1 xyz \, dz \, dy \, dx \): Inner \( \int_0^1 xyz \, dz = xy \cdot \frac{z^2}{2} \big|_0^1 = \frac{xy}{2} \), Middle \( \int_0^2 \frac{xy}{2} \, dy = \frac{x}{2} \cdot y^2 \big|_0^2 = 2x \), Outer \( \int_0^1 2x \, dx = x^2 \big|_0^1 = 1 \).
- \( \int_0^1 \int_0^1 \int_0^2 xyz \, dy \, dz \, dx \): Inner \( \int_0^2 xyz \, dy = xz \cdot y^2 \big|_0^2 = 4xz \), Middle \( \int_0^1 4xz \, dz = 4x \cdot \frac{z^2}{2} \big|_0^1 = 2x \), Outer \( \int_0^1 2x \, dx = 1 \).
Answer: 1 (same both ways).
4) Non-Rectangular Regions 🔍
Triple integrals over non-rectangular regions require defining \( E \) with inequalities (e.g., a sphere \( x^2 + y^2 + z^2 \leq 1 \)). Limits depend on other variables.
Definition 21.3: Non-Rectangular Region
For \( E \) defined by \( a \leq x \leq b \), \( g_1(x) \leq y \leq g_2(x) \), \( h_1(x, y) \leq z \leq h_2(x, y) \), the integral is \( \int_a^b \int_{g_1(x)}^{g_2(x)} \int_{h_1(x, y)}^{h_2(x, y)} f(x, y, z) \, dz \, dy \, dx \).
Example 5: Tetrahedral Region
Evaluate \( \iiint_E 1 \, dV \) where \( E \) is \( 0 \leq x \leq 1 \), \( 0 \leq y \leq 1 - x \), \( 0 \leq z \leq 1 - x - y \).
- \( \int_0^1 \int_0^{1-x} \int_0^{1-x-y} 1 \, dz \, dy \, dx \).
- Innermost: \( \int_0^{1-x-y} 1 \, dz = 1 - x - y \).
- Middle: \( \int_0^{1-x} (1 - x - y) \, dy = [y - xy - \frac{y^2}{2}]_0^{1-x} = 1 - x - (1 - x)^2/2 \).
- Outer: \( \int_0^1 \left(1 - x - \frac{(1 - x)^2}{2}\right) \, dx = \frac{1}{6} \).
Answer: \( \frac{1}{6} \) (volume of tetrahedron).
Example 6: Spherical Region
Compute \( \iiint_E 1 \, dV \) where \( E \) is \( x^2 + y^2 + z^2 \leq 1 \).
- Use spherical coordinates (later topic preview): \( x = \rho \sin\phi \cos\theta \), \( y = \rho \sin\phi \sin\theta \), \( z = \rho \cos\phi \), \( dV = \rho^2 \sin\phi \, d\rho \, d\phi \, d\theta \).
- Limits: \( \rho \) from 0 to 1, \( \phi \) from 0 to \( \pi \), \( \theta \) from 0 to \( 2\pi \).
- \( \int_0^{2\pi} \int_0^{\pi} \int_0^1 \rho^2 \sin\phi \, d\rho \, d\phi \, d\theta = \int_0^{2\pi} d\theta \cdot \int_0^{\pi} \sin\phi \, d\phi \cdot \int_0^1 \rho^2 \, d\rho \).
- \( \int_0^1 \rho^2 \, d\rho = \frac{1}{3} \), \( \int_0^{\pi} \sin\phi \, d\phi = -cos\phi \big|_0^{\pi} = 2 \), \( \int_0^{2\pi} d\theta = 2\pi \).
- Total: \( 2\pi \cdot 2 \cdot \frac{1}{3} = \frac{4\pi}{3} \).
Answer: \( \frac{4\pi}{3} \) (volume of unit sphere).
5) Applications of Triple Integrals 🔍
Triple integrals calculate volume, mass, center of mass, and moments of inertia in 3D. For a density \( \rho(x, y, z) \), mass is \( \iiint_E \rho(x, y, z) \, dV \).
Definition 21.4: Applications
- Volume: \( \iiint_E 1 \, dV \). - Mass: \( \iiint_E \rho(x, y, z) \, dV \). - Center of Mass: \( \bar{x} = \frac{\iiint_E x \rho \, dV}{M} \), similarly for \( \bar{y} \), \( \bar{z} \).
Example 7: Volume of a Solid
Find the volume under \( f(x, y, z) = 8 - 2x - 2y - z \) over \( E = [0, 1] \times [0, 1] \times [0, 1] \).
- \( \iiint_E (8 - 2x - 2y - z) \, dV \).
- \( \int_0^1 \int_0^1 \int_0^1 (8 - 2x - 2y - z) \, dz \, dy \, dx \).
- Innermost: \( \int_0^1 (8 - 2x - 2y - z) \, dz = [8z - 2xz - 2yz - \frac{z^2}{2}]_0^1 = 8 - 2x - 2y - \frac{1}{2} \).
- Middle: \( \int_0^1 (8 - 2x - 2y - \frac{1}{2}) \, dy = [8y - 2xy - y^2 - \frac{y}{2}]_0^1 = 8 - 2x - 1 - \frac{1}{2} = 6.5 - 2x \).
- Outer: \( \int_0^1 (6.5 - 2x) \, dx = [6.5x - x^2]_0^1 = 6.5 - 1 = 5.5 \).
Answer: 5.5 cubic units.
Example 8: Mass with Density
Find the mass of a solid \( E = [0, 1] \times [0, 1] \times [0, 1] \) with density \( \rho(x, y, z) = x + y + z \).
- \( \iiint_E (x + y + z) \, dV \).
- \( \int_0^1 \int_0^1 \int_0^1 (x + y + z) \, dz \, dy \, dx \).
- Innermost: \( \int_0^1 (x + y + z) \, dz = [xz + yz + \frac{z^2}{2}]_0^1 = x + y + \frac{1}{2} \).
- Middle: \( \int_0^1 (x + y + \frac{1}{2}) \, dy = [xy + \frac{y^2}{2} + \frac{y}{2}]_0^1 = x + \frac{1}{2} + \frac{1}{2} = x + 1 \).
- Outer: \( \int_0^1 (x + 1) \, dx = [\frac{x^2}{2} + x]_0^1 = \frac{1}{2} + 1 = 1.5 \).
Answer: Mass = 1.5 units.
6) Practice Questions 🎯
Fundamental Practice Questions 🌱
Instructions: Evaluate the triple integral over the given region. 📚
\( \iiint_E 1 \, dV \) where \( E = [0, 2] \times [0, 1] \times [0, 3] \)
\( \iiint_E x \, dV \) where \( E = [0, 1] \times [0, 2] \times [0, 1] \)
\( \iiint_E (x + y + z) \, dV \) where \( E = [0, 1] \times [0, 1] \times [0, 1] \)
\( \iiint_E z^2 \, dV \) where \( E = [0, 2] \times [0, 1] \times [0, 1] \)
\( \iiint_E \sin(x) \, dV \) where \( E = [0, \pi] \times [0, 1] \times [0, 2] \)
\( \iiint_E x^2 + y^2 + z^2 \, dV \) where \( E = [0, 1] \times [0, 1] \times [0, 1] \)
\( \iiint_E e^z \, dV \) where \( E = [0, 1] \times [0, 2] \times [0, 1] \)
\( \iiint_E xyz \, dV \) where \( E = [0, 1] \times [0, 1] \times [0, 2] \)
\( \iiint_E \cos(x + y + z) \, dV \) where \( E = [0, \pi/2] \times [0, \pi/2] \times [0, \pi/2] \)
\( \iiint_E 1 \, dV \) where \( E = [-1, 1] \times [0, 2] \times [0, 1] \)
\( \iiint_E x^2 y z \, dV \) where \( E = [0, 2] \times [0, 1] \times [0, 3] \)
Challenging Practice Questions 🌟
Instructions: Solve these advanced triple integrals or applications. 🧠
Evaluate \( \iiint_E (x^2 + y^2 + z^2) \, dV \) where \( E \) is \( 0 \leq x \leq 1 \), \( 0 \leq y \leq 1 - x \), \( 0 \leq z \leq 1 - x - y \).
Compute the volume under \( f(x, y, z) = 12 - 3x - 2y - 4z \) over \( E = [0, 2] \times [0, 1] \times [0, 1] \).
Find the mass of a solid \( E = [0, 1] \times [0, 1] \times [0, 1] \) with density \( \rho(x, y, z) = x^2 + y^2 + z^2 \).
Determine \( \iiint_E e^{x+y+z} \, dV \) where \( E \) is \( 0 \leq x \leq 1 \), \( 0 \leq y \leq x \), \( 0 \leq z \leq y \).
Evaluate the volume of the region \( E \) defined by \( x^2 + y^2 + z^2 \leq 9 \) using a triple integral (hint: consider spherical symmetry).
7) Summary & Cheat Sheet 📋
7.1) Triple Integral
\( \iiint_E f(x, y, z) \, dV \) computes volume or properties over 3D region \( E \).
7.2) Iterated Integral
\( \int_a^b \int_c^d \int_e^f f(x, y, z) \, dz \, dy \, dx \), with flexible order.
7.3) Applications
Volume, mass, center of mass via \( \iiint_E f(x, y, z) \, dV \).
You’ve mastered triple integrals! Next, we’ll explore change of variables. 🎉