Calculus 3

Calculus 3 covers advanced topics in multivariable calculus, including partial derivatives, multiple integrals, and vector calculus.

Functions of Several VariablesPartial DerivativesMultiple Integrals
Vector Fields and GradientLine and Surface IntegralsDivergence, Curl, and the Big Theorems
Modeling Weather and ClimateEngineering and Physics ApplicationsEconomics and Resource Management
Visualization and SketchingBreaking Down Problems
Functions of Several...Partial DerivativesMultiple IntegralsVector Fields and Gr...Line and Surface Int...Divergence, Curl, an...Modeling Weather and...Engineering and Phys...Economics and Resour...Visualization and Sk...Breaking Down Proble...
Basic Concepts

Multiple Integrals

Summing Over Areas and Volumes

Multiple integrals allow us to add up infinitely many tiny pieces over areas (double integrals) or volumes (triple integrals). It's like extending the idea of finding area under a curve to higher dimensions.

Double and Triple Integrals

  • Double Integrals (\( \iint \)): Used to find area and accumulated quantities over a region in 2D.
  • Triple Integrals (\( \iiint \)): Used to find volume and accumulated quantities in 3D.

Setting Up the Integral

To compute a double integral, we often integrate with respect to one variable while treating the other as constant, then repeat for the second variable.

Applications

Multiple integrals are used to calculate things like mass, probability, and energy distributed over regions and spaces.

Step-by-Step

  1. Identify the region to integrate over.
  2. Set up the limits for each variable.
  3. Integrate step by step, usually from inside out.

Examples

  • Finding the mass of a thin plate with variable density using a double integral.

  • Calculating the probability over a region with a probability density function.

In a Nutshell

Multiple integrals let us find areas, volumes, and more in higher dimensions.

Next
Vector Fields and Gradient