Double Integral
26 min 3 Examples
- Double Integrals over Rectangles as it relates to Riemann Sums from Calc 1
- Overview of how to approximate the volume Analytically and Geometrically using Riemann Sums
- Example of approximating volume over a square region using lower left sample points
- Example of approximating volume over a square region using upper right sample points
- Example of approximating volume using the Midpoint Rule
Average Value Of A Double Integral
21 min 5 Examples
- Overview of finding the average value in single-variable and multivariable calculus
- Example of finding the average value of a function of two variables
- Overview of the Properties of Double Integrals
- Four examples of evaluating double integrals over rectangles using properties
Iterated Integral
34 min 3 Examples
- Overview of how to evaluate a double integral using Fubini’s Theorem
- Demonstration of how to use Fubini’s Theorem
- Example #1 of writing and evaluating an Iterated Integral of a rectangular region
- Example #2 of writing and evaluating an Iterated Integral of a rectangular region using trig
- Example #3 of writing and evaluating an Iterated Integral of a rectangular region using U-Substitution
Double Integral Over General Regions
1 hr 36 min 9 Examples
- How did we find Area Between Curves in Single-Variable Calculus?
- Example showing the relationship between Single and Double Integrals for finding area
- Example #1 using Iterated Integrals finding Area of a Bounded Region
- Example #2 using Iterated Integrals finding Area over General Region
- Example #3 using Iterated Integrals finding Area over General Region
- Example #4 using Iterated Integrals finding Area over General Region
- Overview and Example of how to Reverse/Switch the Order of Integration
- Example #2 of how to Reverse/Switch the Order of Integration
- Example #3 of how to Reverse/Switch the Order of Integration
- Application Example of Double Integrals over General Regions to find Volume
Double Integral With Polar Coordinates
1 hr 5 min 5 Examples
- Overview and formula for how to calculate double integrals in polar coordinates
- Example #1 of changing to polar coordinates and evaluating the double integral
- Example #2 of changing to polar coordinates and evaluating the double integral using U-Substitution
- Example #3 of changing to polar coordinates and evaluating the double integral using U-Substitution
- Example #4 of evaluating a double integral in polar coordinates using a half-angle identity
- Example #5 of finding the volume of a solid in polar coordinates
Applications Of Double Integrals
1 hr 1 min 4 Examples
- Overview of Mass, Density, Moments, and the Center of Mass for Double Integrals
- Example of how to find the Mass and Center of Mass using Double Integrals
- Example of finding Total Charge given charge density at a point
- Overview of Moments of Inertia (Second Moments)
- Example of finding the Moments of Inertia
- Overview of Probability Density
- Example of finding the Probability Density for a Joint Density Function
Surface Area
38 min 3 Examples
- Overview of Surface Area using Double Integrals
- Example #1 of finding Surface Area of the part of a plane
- Example #2 of finding Surface Area of a cylinder
- Example #3 of finding Surface Area using Polar Coordinates
Triple Integral
44 min 3 Examples
- Overview of Triple Integrals
- Example #1 Evaluating a Triple Integral over a rectangular box
- Example #2 Evaluating a Triple Integral over a general bounded region
- Example #3 Evaluating a Triple Integral over a general bounded region
Triple Integrals in Cylindrical Coordinates
43 min 3 Examples
- Overview and Formula for Triple Integrals in Cylindrical Coordinates
- Example #1 Evaluating a Triple Integral in Cylindrical Coordinates
- Example #2 Evaluating a Triple Integral in Cylindrical Coordinates
- Example #3 Converting a Triple Integral from Rectangular to Cylindrical
Triple Integrals in Spherical Coordinates
37 min 3 Examples
- Review of relationship between Rectangular and Spherical Coordinates
- Formula for Triple Integrals in Spherical Coordinates
- Example #1 Evaluating a Triple Integral in Spherical Coordinates
- Example #2 Evaluating a Triple Integral in Spherical Coordinates
- Example #3 Converting a Triple Integral from Rectangular to Spherical
Change Of Variables
1 hr 6 min 7 Examples
- Overview of Change of Variables and Transformations
- Example #1 Determining a new region by applying a given transformation
- Example #2 Determining a new region by applying a given transformation
- Overview of the Jacobian Transformation
- Example #1 Finding the Jacobian Determinate for two variables
- Example #2 Finding the Jacobian Determinate in polar coordinates
- Example #3 Finding the Jacobian Determinate for three variables
- Overview and Formula for finding the Change of Variables for Multiple Integrals
- Example #1 Evaluating a double integral given an appropriate change of variables
- Example #2 Evaluating a double integral given an appropriate change of variables
Chapter Test
2 hr 24 min 18 Practice Problems
- Determine whether the statements is true or false (Problems #1-4)
- Use a Riemann Sum to approximate the double integral using midpoints or corner points (Problems #5-6)
- Evaluate the Double or Triple Integral (Problems #7-8)
- Calculate the integral by first reversing the order of integration (Problems #9-10)
- Calculate the value of the iterated integral (Problem #11)
- Find the surface area of the portion of the plane (Problem #12)
- Calculate the integral over general regions (Problems #13-14)
- Find the volume of the solid (Problem #15)
- Find the center of mass of the lamina (Problem #16)
- Find the volume using the given transformation (Problem #17)
- Set up the integral in rectangular, cylindrical, and spherical coordinates (Problem #18)
