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When dealing with objects that have variable density, the density changes across different points within the object. In mathematical terms, this is described by a density function, often denoted as \( \rho(x, y, z) \). In our specific problem, the density varies as \( \rho(x, y, z) = 2 + x + y + z \). This means the density at any point

• increases by 1 as you move along the x-axis,
• increases by 1 as you move along the y-axis,
• and again increases by 1 as you move along the z-axis.

This variable density leads to a more complex calculation compared to constant density. Instead of a simple multiplication, we integrate the density function over the volume of the solid to find quantities like mass. Understanding variable density requires recognizing that not all parts of the object contribute equally to the overall mass due to variations in density.

Triple integration is an extension of single integration into three dimensions. It is used when you need to integrate over a region in three-dimensional space. In our problem, the triple integral is expressed as:\[\int_{0}^{1} \int_{0}^{1} \int_{0}^{1} f(x, y, z) \, dx \, dy \, dz\]Where the function \( f(x, y, z) \) is the density function \( \rho(x, y, z) \). This triple integral calculates volume attributes, such as the mass, of a three-dimensional object.
The process follows these steps:

• Evaluate the innermost integral: Typically, the integrals are computed from the innermost to the outermost. For the innermost integral (with respect to \(x\)), treat \(y\) and \(z\) as constants.
• Evaluate the middle integral: Once the innermost integration is done, move onto the next integral (with respect to \(y\)), treating \(z\) as constant.
• Evaluate the outermost integral: Finally, integrate with respect to \(z\).

This method lets us handle complex geometries and variable properties, like density, effectively.

Calculating the mass of an object with variable density involves setting up and evaluating an integral that takes into account the density function over the object's volume. The formula to do so is:\[M = \int \int \int \rho(x, y, z) \, dV\]Where \( \rho(x, y, z) \) is the density function and \( dV \) represents the infinitesimal volume element, often written as \( dx \, dy \, dz \) in Cartesian coordinates.
Here's how the steps unfold in practice:

• Define the limits of integration: Based on the volume the object occupies. In our example, it’s a cube within the first octant, ranging from 0 to 1 for \(x\), \(y\), and \(z\).
• Integrate the density function: Over the defined limits. In our case, we found that the mass \( M = 4 \).

The total mass calculation tells us how heavy the object is overall, even if its density is not constant throughout. It’s crucial in finding the center of mass, which uses the mass as a reference.