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# 3: Multiple Integrals

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Multiple integrals are a generalization of the definite integral to functions of more than one variable.

• 3.1: Double and Iterated Integrals Over Rectangles
Thus, we can conclude that the integral is the function of accumulation as it accumulates infinite number of strips in a certain domain to calculate the area. Similarly, the double integral is also a function of accumulation. It accumulates infinite number of small 3D strips to calculate the volume of 3D objects.
• 3.2: Area by Double Integration
In this section, we will learn to calculate the area of a bounded region using double integrals, and using these calculations we can find the average value of a function of two variables.
• 3.3: Double Integrals Over General Regions
In 15.1, we notice that all bases of the objects are rectangular. In 15.2, the area under these objects are nonrectangular. However, the method of accumulation still works.
• 3.4: Double Integrals in Polar Form
If the domain has the characteristics of a circle or cardioid, then it is much easier to solve the integral using polar coordinates.
• 3.5: Triple Integrals in Rectangular Coordinates
Just as a single integral has a domain of one-dimension (a line) and a double integral a domain of two-dimension (an area), a triple integral has a domain of three-dimension (a volume). Furthermore, as a single integral produces a value of 2D and a double integral a value of 3D, a triple integral produces a value of higher dimension beyond 3D, namely 4D.
• 3.6: Triple Integrals in Cylindrical and Spherical Coordinates
Sometimes, you may end up having to calculate the volume of shapes that have cylindrical, conical, or spherical shapes and rather than evaluating such triple integrals in Cartesian coordinates, you can simplify the integrals by transforming the coordinates to cylindrical or spherical coordinates. For this topic, we will learn how to do such transformations then evaluate the triple integrals.
• 3.7: Moments and Centers of Mass
This section shows how to calculate the masses and moments of two- and three- dimensional objects in Cartesian (x,y,z) coordinates.
• 3.8: Jacobians
The goal for this section is to be able to find the "extra factor" for a more general transformation. We call this "extra factor" the Jacobian of the transformation. We can find it by taking the determinant of the two by two matrix of partial derivatives.
• 3.9: Substitutions in Multiple Integrals
This section discusses the translation of a graph from the xy Cartesian plane to the uv Cartesian plane and defines the Jacobian. The Jacobian measures how much the volume at a certain point changes when being transformed from one coordinate system to another.