Applications of Double Integrals
We have already considered applications of double integrals, in fact we actually started with an application: finding the mass of a plate of variable density. We have also found volumes of solids.
We consider here further applications. These are based on formulas which we first obtain from localized cases, say for a particle. We then extend them to two dimensional regions, by summing and taking limits (i.e., integrating) just like we did before.
We recall the following: Suppose we have a particle of mass M positioned at . Then:
= 1st moment with respect to the x-axis = ;
= 1st moment with respect to the y-axis = ;
= 2nd moment (moment of inertia) with respect to the x-axis =
= 2nd moment (moment of inertia) with respect to the y-axis =
= 2nd moment (moment of inertia) about the origin (polar moment) = .
The total mass is represented by
is the density at the , where
point (x, y)
With: Then the center of mass of R is
Example:
A rectangular metal sheet with 2 <x <5 and 0 <y <3 has density function
Set up the double integral that gives the mass of the metal sheet.
Solution:
We just have the integral
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