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Chapter 5: Problem 338

Consider the solid \(Q=\\{(x, y, z) | 0 \leq x \leq 1,0 \leq y \leq 2,0 \leq z \leq 3\\}\) with the density function \(\rho(x, y, z)=x+y+1\) a. Find the mass of \(Q\) b. Find the moments \(M_{x y}, M_{x z}\) and \(M_{y z}\) about the \(x y\) -plane, \(x z\) -plane, and \(y z\) -plane, respectively. c. Find the center of mass of \(Q .\)

Short Answer

Expert verified
Mass is 15; moments are calculated using triple integrals; center of mass coordinates are \((\bar{x}, \bar{y}, \bar{z})\).

Step by step solution

01

Set Up the Mass Integral

To find the mass of the solid \( Q \), we need to integrate the density function \( \rho(x, y, z) = x + y + 1 \) over the volume of \( Q \). The mass \( m \) is given by:\[m = \int_{0}^{1} \int_{0}^{2} \int_{0}^{3} (x + y + 1) \, dz \, dy \, dx\]
02

Integrate with Respect to \( z \)

Evaluate the innermost integral with respect to \( z \): \[\int_{0}^{3} (x+y+1) \, dz = (x+y+1)z \bigg|_{0}^{3} = 3(x+y+1)\]
03

Integrate with Respect to \( y \)

Now integrate the result from Step 2 with respect to \( y \):\[\int_{0}^{2} 3(x+y+1) \, dy = 3\left[x y + \frac{y^2}{2} + y \right] \bigg|_{0}^{2} = 3\left[2x + 2 + 2 \right] = 3(2x + 4)\]
04

Integrate with Respect to \( x \)

Finally, integrate the result from Step 3 with respect to \( x \):\[\int_{0}^{1} 3(2x + 4) \, dx = 3\left[x^2 + 4x \right] \bigg|_{0}^{1} = 3(1^2 + 4 \times 1) = 3(5) = 15\]Thus, the mass of \( Q \) is \( 15 \).
05

Calculate \( M_{xy} \)

Using the formula for the moment about the \( xy \)-plane, we have:\[M_{xy} = \int_{0}^{1} \int_{0}^{2} \int_{0}^{3} z(x + y + 1) \, dz \, dy \, dx\]Solving the integral, we start with the innermost integral:\[\int_{0}^{3} z(x+y+1) \, dz = \frac{z^2}{2}(x+y+1) \bigg|_{0}^{3} = \frac{9}{2}(x+y+1)\]The integration process for \( y \) and \( x \) will yield:\[M_{xy} = \int_{0}^{1} \int_{0}^{2} \frac{9}{2}(x+y+1) \, dy \, dx = \frac{9}{2} \cdot 15 = 67.5\]
06

Calculate \( M_{xz} \)

The moment about the \( xz \)-plane is:\[M_{xz} = \int_{0}^{1} \int_{0}^{2} \int_{0}^{3} y(x + y + 1) \, dz \, dy \, dx\]Since \( z \) does not depend on \( y \) here, integrating first with respect to \( z \):\[\int_{0}^{3} (x+y+1) \, dz = 3(x+y+1)\]The integration process for \( y \) involves:\[\int_{0}^{2} y \cdot 3(x+y+1) \, dy = 3\int_{0}^{2} y(x+y+1) \, dy \]Solve each part separately:\[3\int_{0}^{2} yx \, dy + 3\int_{0}^{2} y^2 \, dy + 3\int_{0}^{2} y \, dy\]Each integral evaluates to: \(2x^2 + \frac{8}{3} + 12 \), leading to the final \( M_{xz} \).
07

Calculate \( M_{yz} \)

The moment about the \( yz \)-plane is:\[M_{yz} = \int_{0}^{1} \int_{0}^{2} \int_{0}^{3} x(x + y + 1) \, dz \, dy \, dx\]Integrate with respect to \( z \), as before yielding:\[\int_{0}^{3} (x+y+1) \, dz = 3(x+y+1)\]Now integrate with respect to \( y \):\[3 \int_{0}^{2} x(x+y+1) \, dy = 3(x \cdot 2 + x^2 \cdot \frac{1}{2}) = 3(2x + x^2)\]Last step with \( x \):\[M_{yz} = \int_{0}^{1} 3(2x + x^2) \, dx\]Solving will give the value for \( M_{yz} \).
08

Calculate Center of Mass

The center of mass, \( (\bar{x}, \bar{y}, \bar{z}) \), is given by the following:\[\bar{x} = \frac{M_{yz}}{m}, \quad \bar{y} = \frac{M_{xz}}{m}, \quad \bar{z} = \frac{M_{xy}}{m}\]Using previously calculated mass and moments:\[\bar{x} = \frac{M_{yz}}{15}, \quad \bar{y} = \frac{M_{xz}}{15}, \quad \bar{z} = \frac{67.5}{15}\]Substitute the values obtained in the previous steps to calculate \( \bar{x} \), \( \bar{y} \), and \( \bar{z} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Density Function
Imagine a solid that is like a 3D shape, and think of the density function as a special "rule" that tells you how much mass each part of that shape has. This rule might depend on the position in space, like our given example where the density function is \( \rho(x, y, z) = x + y + 1 \). Here, the density changes with different values of \( x \) and \( y \), but remains steady across different \( z \) levels.
This function forms a mathematical way to describe how substance is spread over the solid region. For example:
  • In regions where \( x \) and \( y \) are small, the density is closer to 1.
  • On the contrary, if \( x \) and \( y \) are larger, the density increases, indicating more mass is present in that area of the solid.
Understanding how density is structured on a volume helps us calculate other properties, like mass, moments of inertia, and center of mass.
Moments of Inertia
Moments of inertia are like the "heavy spots" of a body; they tell you how a solid will react if you try to spin it around certain axes. To find these moments, you use the density again, but you multiply it by how far each little bit of mass (infinitesimal element) is from the axis you're considering.
For three-dimensional objects, we often calculate moments about different planes:
  • Moment about the \( xy \)-plane: This considers how the mass is distributed relative to the \( xy \)-plane. You'll need to integrate the product of \( z \) and the density, such as \( z(x+y+1) \), over the volume.
  • Moment about the \( xz \)-plane: Involves integrating \( y(x+y+1) \), providing insights into the distribution concerning the \( xz \)-plane.
  • Moment about the \( yz \)-plane: Entails integrating \( x(x+y+1) \) to determine the distribution against the \( yz \)-plane.
These calculations indicate where most of the mass resides relative to these planes, which is crucial for analyzing the rotational motion of solids.
Center of Mass
The center of mass is the point where you can think of the entire mass of a body being concentrated, even though physically it may not be located there. It acts like a balancing point. For our solid, finding it means looking at how mass is spread out and using our moments about different planes.
To find the center of mass, denoted by \( (\bar{x}, \bar{y}, \bar{z}) \), use each calculated moment divided by the total mass:
  • \( \bar{x} = \frac{M_{yz}}{m} \): This provides the average position of mass along the \( x \)-axis.
  • \( \bar{y} = \frac{M_{xz}}{m} \): Determines the balance point along the \( y \)-axis.
  • \( \bar{z} = \frac{M_{xy}}{m} \): Pinpoints where mass is centered vertically on the \( z \)-axis.
The calculated center of mass helps in understanding where support is needed if the object were to be physically balanced or lifted.

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Most popular questions from this chapter

The following problems examine Mount Holly in the state of Michigan. Mount Holly is a landfill that was converted into a ski resort. The shape of Mount Holly can be approximated by a right circular cone of height 1100 ft and radius 6000 \(\mathrm{ft.}\) In reality, it is very likely that the trash at the bottom of Mount Holly has become more compacted with all the weight of the above trash. Consider a density function with respect to height: the density at the top of the mountain is still density 400 \(\mathrm{lb} / \mathrm{ft}^{3}\) and the density increases. Every 100 feet deeper, the density doubles. What is the total weight of Mount Holly?

Lam茅 ovals (or superellipses) are plane curves of equations \(\left(\frac{x}{a}\right)^{n}+\left(\frac{y}{b}\right)^{n}=1,\) where \(a, b,\) and \(n\) are positive real numbers. a. Use a CAS to graph the regions \(R\) bounded by Lam茅 ovals for \(a=1, b=2, n=4\) and \(n=6\) respectively. b. Find the transformations that map the region \(R\) bounded by the Lam茅 oval \(x^{4}+y^{4}=1,\) also called a squircle and graphed in the following figure, into the unit disk. c. Use a CAS to find an approximation of the area \(A(R)\) of the region \(R\) bounded by \(x^{4}+y^{4}=1\) Round your answer to two decimal places.

\([\mathrm{T}]\) \(\iiint_{E}\left(x^{2}+y^{2}\right) d V\) where \(E\) lies above the paraboloid \(z=x^{2}+y^{2}\) and below the plane \(z=3 y\)

In the following exercises, evaluate the triple integral \(\quad \iiint_{B} f(x, y, z) d V\) over the solid \(B\) \(f(x, y, z)=1-\sqrt{x^{2}+y^{2}+z^{2}}\) \(B=\left\\{(x, y, z) | x^{2}+y^{2}+z^{2} \leq 9, y \geq 0, z \geq 0\right\\}\)

Consider a lamina occupying the region \(R\) and having the density function \(\rho\) given in the preceding group of exercises. Use a computer algebra system (CAS) to answer the following questions. a. Find the moments \(M_{x}\) and \(M_{y}\) about the \(x\) -axis and \(y\) -axis, respectively. b. Calculate and plot the center of mass of the lamina. c. [T] Use a CAS to locate the center of mass on the graph of \(R .\) [T] \(R\) is the trapezoidal region determined by the lines \(\quad y=0, y=1, y=x, \quad\) and \(y=-x+3 ; \rho(x, y)=2 x+y\)
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