Much like double integrals, triple integrals are integrals of a function of multiple variables but instead of just 2 variables it has 3.
A triple integral over the solid region
Where
Note:
As seen above the inner two integrals must have their bounds written as functions of the outer variables, simply using the bounds is not sufficient.
#flag-review below:
In a triple integral the bounds define the volume of the solid, where as in double integrals they only defined the projection of the solid onto one of the axis planes (xy, xz, and yz-planes). So now that we already have volume covered in our equation this allows us to use a separate function in for
Think about a room that is 10 feet wide, 10 feet deep, and 20 feet tall. You want to find out how much oxygen is in this room but need the bottom of the room the air is more dense and at the top there is almost no air. The density of the air in the room is given by
See Multivariable Calculus Notes - Chapter 15 - Sections 6-8 for more information.
Like double integrals you first solve the inner integrals and then work your way to the outer integrals. You should have to solve 3 integrals total.
See Multivariable Calculus Notes - Chapter 15 - Sections 6-8 for examples.
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Spherical and cylindrical coordinate systems can be used with triple integrals to simplify the evaluation process in some equations. It may be useful to employ trigonometry identities to simplify equations when using these methods.
Best used for solids that are spherical in nature, and generally centered close to the origin.
One uses the spherical coordinate system by first substituting the
In addition to the added functions in the integration terms:
Notice the added
Best used for solids that are cylindrical, or round in two of the three axis, where the
One uses cylindrical coordinates by substituting the
In addition to added an
One notices that these substitutions are very similar to polar coordinates and indeed finds that the only difference is the addition of the