Question

Find the centroid of the solid region bounded by the graphs of the equations. Use a computer algebra system to evaluate the triple integrals. (Assume uniform density and find the center of mass.) $$ z=\frac{1}{y^{2}+1}, z=0, x=-2, x=2, y=0, y=1 $$

Short answer

The centroid of the solid region (\(\bar{x}, \bar{y}, \bar{z}\)) can be calculated by evaluating the given triple integrals using a computer algebra system due to the complexity. These values essentially represent the center of mass of the solid region.

Step-by-step solution

Formulas for Centroid

The centroid of a solid region in three dimensions is given by \((\bar{x}, \bar{y}, \bar{z})\) where \(\bar{x} = \frac{1}{V} \int_D xdV\), \(\bar{y} = \frac{1}{V} \int_D ydV\), \(\bar{z} = \frac{1}{V} \int_D zdV\) and \(V = \int_D dV\n) is the volume of the solid region. \(D\) represents the solid region under consideration.

Formulate x, y, and z Boundary Limits for Integration

The limits for \(x\), \(y\), and \(z\) can be obtained from the given equations. The limits for \(x\) are -2 and 2. The limits for \(y\) are 0 and 1. And the limits for \(z\) are \(0\) and \(\frac{1}{y^{2}+1}\).

Evaluate the Volume Integral

Volume \(V\) can be formulated as a triple integral with respect to \(x, y, z\). \n\(V = \int_{-2}^{2} \int_{0}^{1} \int_{0}^{\frac{1}{y^{2}+1}} dzdydx\). This integral could be evaluated using a computer algebra system.

Evaluate the Integral for Centroid Coordinate

The values of \(\bar{x}, \bar{y}, \bar{z}\) can be computed with their respective triple integrals. \n\(\bar{x} = \frac{1}{V} \int_{-2}^{2} \int_{0}^{1} \int_{0}^{\frac{1}{y^{2}+1}} x dzdydx\), \(\bar{y} = \frac{1}{V} \int_{-2}^{2} \int_{0}^{1} \int_{0}^{\frac{1}{y^{2}+1}} y dzdydx\), \(\bar{z} = \frac{1}{V} \int_{-2}^{2} \int_{0}^{1} \int_{0}^{\frac{1}{y^{2}+1}} z dzdydx\). Again, these integrals can be evaluated using a computer algebra system due to the complexity of the bounds.

Key concepts

Triple Integrals

Triple integrals are at the heart of calculating various properties of three-dimensional spaces, especially in the context of physics and engineering. Imagine slicing a three-dimensional object into infinitesimally thin pieces, examining each piece, and then summing up their contributions to get a complete picture. That's essentially what a triple integral does.
When finding the centroid of a solid region with uniform density, these integrals sum up the moments, or 'weighted averages,' across the entire volume of a solid. In simpler terms, they help us determine the average location of all the mass in the solid.

• How do they work? Triple integrals extend the concept of double integrals by adding another dimension.
• Importance? They allow for the calculation of volume as well as center of mass, among other things.
• Application? In the provided exercise, triple integrals are used to calculate the volume and the coordinates of the centroid.

Each integral takes into account a different coordinate — x, y, or z — and integrates a function over the region defined by our limits, in this case, the solid defined by the given equations.

Computer Algebra System

A Computer Algebra System (CAS) is an incredibly powerful tool for mathematicians, engineers, and scientists, capable of manipulating mathematical expressions and solving complex equations that would be time-consuming or impractical to solve by hand.

• Purpose? It allows for the symbolic manipulation of algebraic expressions, rather than just number crunching.
• Benefits? Fast, accurate computations and the ability to handle very complex problems that are often involved in higher mathematics.
• In our exercise? A CAS would take the complex bounds and functions from the triple integrals and solve them, providing precise values for the volume and centroid coordinates without manual calculation errors.

In educational settings, a CAS is a valuable assistant, allowing students to check their work or understand complex concepts by visualizing and manipulating them interactively.

Volume of Solid

When we talk about the volume of a solid in mathematics, we refer to the amount of space that the solid occupies. Determining the volume is essential not just for real-world applications like construction and manufacturing, but also for further calculations in calculus, such as finding the center of mass.

• Critical step? To find the centroid, we first need to know the total volume, since the centroid is effectively the 'balance point' of the volume.
• Triple integrals and volume? The triple integral calculated in the exercise for the volume is the foundation for finding the centroid coordinates.
• Real-world analogy? Think of the volume as a measure of how much water the solid could hold if it were hollow — this property is fundamental to understanding the space within boundaries.

In the provided exercise, the volume was determined by setting up a triple integral in the order of dz, dy, dx, taking the solid's boundaries into account.

Center of Mass

The center of mass of an object is a point that represents the average location of all the mass in the object. In the context of physics, it's the point at which you could balance the object perfectly if it were possible to support it there.

• Why is it important? It's a critical concept in engineering and physics because it affects how objects behave under forces, like gravity.
• For uniform density solids, center of mass and centroid coincide, simplifying our calculations.
• Connections to the exercise? The exercise involves calculating the center of mass for a solid with uniform density, making the centroid its geometric equivalent.

By calculating the centroids along each axis using triple integrals, as shown in the step-by-step solution, we can pinpoint the center of mass for the three-dimensional solid in question.