Question

Set up a triple integral that gives the moment of inertia about the \(z\) -axis of the solid region \(Q\) of density \(\rho .\) $$ \begin{array}{l} Q=\\{(x, y, z):-1 \leq x \leq 1,-1 \leq y \leq 1,0 \leq z \leq 1-x\\} \\ \rho=\sqrt{x^{2}+y^{2}+z^{2}} \end{array} $$

Short answer

The triple integral for the moment of inertia about the z-axis of the solid region Q with the given density \(\rho\) is \(I = \int_{-1}^{1} \int_{-1}^{1} \int_{0}^{1-x^2} z (x^2 + y^2 + z^2) dz dx dy\).

Step-by-step solution

Setting Up the Integral

First, the formula to apply is \(I = \int \int \int_V \rho R^2 dV\). Our object has a density \(\rho = \sqrt{x^{2}+y^{2}+z^{2}}\). The distance to the z-axis in this case is \(R^2 = x^2 + y^2\). Therefore, the function under the integral becomes \(\rho R^2 = x^2+y^2+r^2\). With dV taken as \(dz dx dy\), this gives us the integral \(I = \int \int \int_V z (x^2 + y^2 + z^2) dz dx dy\).

Determining The Limts of Integration

The region for integration is given as \(Q = \{(x,y,z): -1 \leq x \leq 1, -1 \leq y \leq 1, 0 \leq z \leq 1-x^2\}\). Therefore, the triples integral for our moment of inertia becomes \(I = \int_{-1}^{1} \int_{-1}^{1} \int_{0}^{1-x^2} z (x^2 + y^2 + z^2) dz dx dy\). This triple integral sums over volume elements within the given region, each volume element weighted by the respective moment of inertia.

Key concepts

Triple Integral

The concept of a triple integral is central to many applications of calculus in three dimensions, especially in physics and engineering. To envision a triple integral, imagine you are trying to calculate the volume under a three-dimensional surface within a certain region of space. However, triple integrals can also be used to compute more complex quantities, such as mass, charge, and, as in our exercise, the moment of inertia.

When we set up a triple integral, we stack an infinite number of infinitesimally thin slices upon each other. Each 'slice' is represented by a double integral, and we integrate these 'slices' over the third dimension to find the triple integral. For instance, to find the volume of a solid region, we integrate a function that represents '1' over the space of the solid. In more complex scenarios, the function might represent density, temperature, or another scalar quantity that varies throughout the space. In our exercise, the triple integral is used to calculate the moment of inertia, where the integrand function includes both the solid's density and the square of the distance to the axis of rotation.

Moment of Inertia

The moment of inertia, often represented as 'I', is a quantity that represents how difficult it is to change the rotational motion of an object. It depends on the mass distribution of the object as well as the axis about which it rotates. Specifically, the moment of inertia is the sum of the products of the mass of each infinitesimal element of the body and the square of its distance from the axis of rotation.

In our context, we're concerned with a continuous distribution of mass throughout a solid region, so instead of summing discrete masses, we integrate. For a solid with variable density, the moment of inertia equation involves the density function (in our case \( \rho = \sqrt{x^2+y^2+z^2} \) ) and feels like a weighted sum where every infinitesimal mass (given by density times volume element) contributes to the overall moment.

Solid Region Density

In physics and engineering problems, we often encounter solid objects that have a non-uniform distribution of mass, which means their density varies throughout the solid. The density function, represented by \( \rho \), expresses how mass is spread throughout a region. It's used to compute physical properties like mass, center of mass, or, as in this exercise, the moment of inertia.

Integral calculus comes into play because we're typically interested in the total effect of this variable density throughout a solid region. The triple integral allows us to add up the infinitesimal contributions from each point within the solid, each contributing according to its local density and position relative to the quantity of interest.

Limits of Integration

The limits of integration in multiple integrals define the boundaries of the region over which we're integrating. Setting up these limits correctly is crucial; they will dictate the 'shape' of the space we are considering. In the case of a triple integral, we have a three-dimensional region, and each set of limits corresponds to one dimension within that region: typically x, y, and z.

When we look at our specific exercise, the region is a solid bounded by planes and curves in three-dimensional space. The limits for each variable give us the 'start' and 'end' points for integration along each axis. Understanding the geometric shape of the region is fundamental, as it helps us visualize and, therefore, set up the integral limits correctly. In our solution, careful attention was given to how the solid Q is defined, and this allowed us to determine the limits for x, y, and z and thereby establish the correct bounds for the integral.