Multivariable Calculus 2: Integrals

Variables are all around us: temperature, altitude, location, profit, color, and countless others. Multivariable Calculus is the tool of choice to shed light on complex relationships between 2, 3, or hundreds of variables simultaneously. Some of the multivariable questions considered in this course include:

• How can one quantify the efficiency of a power plant?
• How much snow can a roof safely hold?
• Over the next 100 years, how high will continually melting icebergs raise sea levels?

The key tool for answering each of these questions is multivariable integration.

In this course, you will learn how to set up, solve, and interpret many types of multivariable integrals:

• double integrals of scalar functions in any coordinate system,
• line integrals of scalar and vector-valued functions,
• triple integrals in cartesian, cylindrical, and spherical coordinates.

Physical applications will be highlighted, including the use of integrals to compute the work done by a force field, or the flux caused by a velocity field.

Finally, you will learn powerful tools for simplifying integral computations, including the Fundamental Theorem of Line Integrals and Green’s Theorem.

Unit 1: Double Integrals

• Double integrals
• Polar coordinates
• Applications:
  • Average value,
  • Center of mass,
  • Moment of inertia
  • Probability
• Change of variables

Unit 2: Introduction to Line Integrals

• Arc length and line integrals
• Force fields and work
• Conservative fields and path independence
• Fundamental Theorem for line integrals

Unit 3: Line Integrals and Green’s Theorem

• Gradient fields and potential functions
• Green’s Theorem
• Flux and Green’s Theorem in normal form
• Simply connected regions

Unit 4: Triple Integrals

• Volume integrals
• Cylindrical coordinates
• Spherical Coordinates
• Review all types of integrals