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Total Mass of a Region in the Plane
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) Text The mass of an object is calculated by integrating its density function over the region of the domain that it occupies.
For a density function ρ(x,y) of two variables and an object in the domain of ρ occupying a region R, the mass is
∫∫Rρ(x,y) dA.
Demos
Density Functions and Mass
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The "Density: rho(x,y)" window shows the density as a color gradient, where cyan indicates rho=0, yellow rho=1, and red rho=2.
The "Mass" window illustrates the mass of the object as the volume underneath the function graph of the density function rho.
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Examples The Mass of an Object with Density @{rho}(x,y) = 4x + 8y@
What is the mass for an object with density ρ(x,y) = 4x + 8y defined over a triangular region with vertices (0,0), (2,1), and (1,0)?
This region can also be defined using the intervals 0 {leq} x {leq} 2 and 0 {leq} y {leq} 0.5x. Using the technique of iterated integration for a region bounded by a function graph of x gives
\begin{array}{rcl}
m = \int_R\int \rho(x,y) dA & = & \int_0^2 \int_0^{.5x} 4x+8y dydx \\
& = & \int_0^2 \left[4xy+4y^2\right]_0^{.5x} dx \\
& = & \int_0^2 3x^2 dx \\
& = & \left[x^3\right]_0^2 = 8
\end{array}
Note that the units of measurement of this value would be determined by the units of the ρ function.
Exercises Will the double integral of rho over this square region find the mass of a three-dimensional object? Explain. Is there a way to modify the double integral to find the mass of a cubic region R determined by 0 {leq} x {leq} 1, 0 {leq} y {leq} 1, 0 {leq} z {leq} 1 using the same density function ρ(x,y)?
Consider a density function of three variables. What would the triple integral of the density function represent? Could the mass be visualized as the volume underneath the function graph of the density function? What is the dimension of the space in which the function graph would exist?
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