Labware - MA35 Multivariable Calculus - Three Variable Calculus
 MA35 Labs 3 » Three Variable Calculus Contents3.4 Integration 3.4.1 Total Mass of a Region in 3-Space 3.4.5 Change of Variables 3.4.6 Center of Mass 3.4.7 Moment of Intertia Search

Total Mass of a Region in 3-Space

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In order to understand how integration works in 3-space, consider the problem of finding the total mass of some region.

If we know density as a function f of x, y, and z, then we can approximate the mass of a tiny box of volume dV located at (x0, y0, z0) as the product f(x0, y0, z0)dV. We can then find the total mass by adding up these the masses of these tiny volumes.

If all three dimensions of the volumes approach 0, the summation becomes a triple integral.

Demos

 Total Mass of a Cubical Region  This demo shows a density function f(x,y,z) over the domain -1 {leq} x, y, z {leq} 1 and computes the total mass over the domain. The function f is initially set to f(x,y,z) = x2 - y2 + z2. Exercises

• How could the total mass method be used as a way to understand integration for functions of one or two variables?