Triple integrals expand on double integrals, letting us calculate volumes and other properties of 3D regions. We'll learn how to set up and evaluate these integrals over both rectangular and more complex general regions.

We'll also explore different coordinate systems like cylindrical and spherical, which can simplify calculations for certain shapes. Understanding how to choose the right system and set up the integral is key to solving these problems efficiently.

Rectangular and General Regions

Defining Rectangular and General Regions

• Rectangular region defined by constant limits of integration
  • Can be evaluated as an iterated integral
  • Limits of integration are constants (a, b, c, d, e, f)
  • Example: $\int_{a}^{b} \int_{c}^{d} \int_{e}^{f} f(x,y,z) , dz , dy , dx$
• General region has variable limits of integration
  • Limits of integration are functions of other variables
  • Requires careful consideration of the order of integration
  • Example: $\int_{a}^{b} \int_{g(x)}^{h(x)} \int_{p(x,y)}^{q(x,y)} f(x,y,z) , dz , dy , dx$

Classifying General Regions

• Type I region has limits of integration for z that depend on both x and y
  • Requires integrating with respect to z first
  • Example: $\int_{a}^{b} \int_{c}^{d} \int_{p(x,y)}^{q(x,y)} f(x,y,z) , dz , dy , dx$
• Type II region has limits of integration for y that depend on x
  • Requires integrating with respect to y second
  • Example: $\int_{a}^{b} \int_{g(x)}^{h(x)} \int_{e}^{f} f(x,y,z) , dz , dy , dx$
• Type III region has constant limits of integration for z
  • Can integrate with respect to z last
  • Example: $\int_{a}^{b} \int_{c}^{d} \int_{e}^{f} f(x,y,z) , dz , dy , dx$

Coordinate Systems

Cylindrical Coordinates

• Cylindrical coordinates $(r, \theta, z)$ useful for regions with cylindrical symmetry
  • $r$ represents the distance from the z-axis in the xy-plane
  • $\theta$ represents the angle in the xy-plane measured counterclockwise from the positive x-axis
  • $z$ represents the vertical distance along the z-axis
• Conversion from rectangular to cylindrical coordinates:
  • $x = r \cos \theta$
  • $y = r \sin \theta$
  • $z = z$
• Volume element in cylindrical coordinates: $dV = r , dr , d\theta , dz$

Spherical Coordinates

• Spherical coordinates $(\rho, \theta, \phi)$ useful for regions with spherical symmetry
  • $\rho$ represents the distance from the origin
  • $\theta$ represents the angle in the xy-plane measured counterclockwise from the positive x-axis
  • $\phi$ represents the angle measured from the positive z-axis
• Conversion from rectangular to spherical coordinates:
  • $x = \rho \sin \phi \cos \theta$
  • $y = \rho \sin \phi \sin \theta$
  • $z = \rho \cos \phi$
• Volume element in spherical coordinates: $dV = \rho^2 \sin \phi , d\rho , d\theta , d\phi$

Integration Techniques

Jacobian and Change of Variables

• Jacobian represents the determinant of the Jacobian matrix
  • Jacobian matrix contains partial derivatives of the coordinate transformation
  • Jacobian accounts for the change in volume element when transforming coordinates
  • Example: Jacobian for cylindrical coordinates is $r$
• Change of variables theorem allows for transforming integrals to different coordinate systems
  • Requires multiplying the integrand by the absolute value of the Jacobian
  • Simplifies the integral by exploiting symmetry or simplifying the region of integration
  • Example: $\iiint_D f(x,y,z) , dV = \iiint_E f(r \cos \theta, r \sin \theta, z) , |r| , dr , d\theta , dz$

Determining Bounds of Integration

• Bounds of integration depend on the region of integration and the coordinate system
  • Rectangular coordinates: Determine the limits for x, y, and z based on the region
  • Cylindrical coordinates: Determine the limits for r, $\theta$, and z based on the region
  • Spherical coordinates: Determine the limits for $\rho$, $\theta$, and $\phi$ based on the region
• Visualizing the region and sketching its boundaries can help determine the limits
  • Consider the shape of the region and how it relates to the chosen coordinate system
  • Example: For a sphere of radius a centered at the origin in spherical coordinates:
    • $0 \leq \rho \leq a$
    • $0 \leq \theta \leq 2\pi$
    • $0 \leq \phi \leq \pi$