Problem 24 - Thomas Calculus

Question

Center of mass \(A\) solid of constant density is bounded below by the plane \(z=0,\) on the sides by the elliptical cylinder \(x^{2}+4 y^{2}=4,\) and above by the plane \(z=2-x\) (see the accompanying figure). a. Find \(\overline{x}\) and \(\overline{y}\) b. Evaluate the integral $$ M_{x y}=\int_{-2}^{2} \int_{-(1 / 2) \sqrt{4-x^{2}}}^{(1 / 2) \sqrt{4-x^{2}}} \int_{0}^{2-x} z d z d y d x $$ using integral tables to carry out the final integration with respect to \(x .\) Then divide \(M_{x y}\) by \(M\) to verify that \(\overline{z}=5 / 4\)

Step-by-Step Solution

Answer

\(\overline{x} = 0\), \(\overline{y} = 0\), and \(\overline{z} = \frac{5}{4}\).

Step 1: Understand the Geometry

The solid is bounded by the elliptical cylinder \(x^2 + 4y^2 = 4\) and lies below the plane \(z = 2 - x\). The region for \(z\) is from 0 to \(2-x\). We need to find the center of mass, starting with \(\overline{x}\) and \(\overline{y}\).

Step 2: Symmetry Analysis for \(\overline{x}\) and \(\overline{y}\)

Due to the symmetry of the elliptical region about the coordinate axes, the center of mass lies on the \(z\)-axis. Thus, \(\overline{x} = 0\) and \(\overline{y} = 0\).

Step 3: Set Up the Triple Integral for \(M_{xy}\)

The integral for \(M_{xy}\) is defined as \(\int_{-2}^{2} \int_{-(1/2)\sqrt{4-x^2}}^{(1/2)\sqrt{4-x^2}} \int_{0}^{2-x} z \, dz \, dy \, dx\). Evaluate it to find the mass moment with respect to the \(xy\)-plane.

Step 4: Evaluate the Inner Integral (\(z\)-direction)

Integrate \(z\) from 0 to \(2-x\): \[\int_{0}^{2-x} z \, dz = \left[ \frac{z^2}{2} \right]_0^{2-x} = \frac{(2-x)^2}{2}\]

Step 5: Evaluate the Middle Integral (\(y\)-direction)

Now, integrate the result from the \(z\) integral over \(y = -(1/2)\sqrt{4-x^2}\) to \(y = (1/2)\sqrt{4-x^2}\):\[\int_{-(1/2)\sqrt{4-x^2}}^{(1/2)\sqrt{4-x^2}} \frac{(2-x)^2}{2} \, dy = \frac{(2-x)^2}{2} \cdot \sqrt{4-x^2}\]

Step 6: Evaluate the Outer Integral (\(x\)-direction)

Finally, integrate with respect to \(x\) from -2 to 2. This resolves to:\[M_{xy} = \int_{-2}^{2} \frac{(2-x)^2}{2} \cdot \sqrt{4-x^2} \, dx\]Use a table of integrals to solve, or numerical integration if tables are not available.

Step 7: Calculate Total Mass \(M\)

Compute the total mass \(M\) of the solid:\[M = \int_{-2}^2 \int_{-(1/2)\sqrt{4-x^2}}^{(1/2)\sqrt{4-x^2}} \int_{0}^{2-x} 1 \, dz \, dy \, dx\]This evaluates similar to previous steps but with a simpler function under integration.

Step 8: Find \(\overline{z}\)

The \(\overline{z}\) is given by \(\overline{z} = \frac{M_{xy}}{M}\). Perform the division from previous calculations to find \(\overline{z} = \frac{5}{4}\).

Key Concepts

Elliptical Cylinder

An elliptical cylinder is a three-dimensional shape generated by extending an ellipse along a straight line, forming cylindrical geometry. In this problem, the elliptical cylinder is described by the equation \(x^2 + 4y^2 = 4\). This equation represents the boundary in the \(xy\)-plane. The cross-section of this cylinder is an ellipse and does not vary along the \(z\)-direction.
To interpret the cylinder's boundaries:

• The coefficient of \(x^2\) is \(1\), signifying that the total width of the ellipse along the \(x\)-axis is \(2\), from \(-2\) to \(2\).
• The coefficient of \(4y^2\) gives the height along the \(y\)-axis as \(1\), from \(-1\) to \(1\), simplified from multiplying by the semi-major axis \(2\), hence expressed as \((1/2)\sqrt{4 - x^2}\) within the integral limits.

This region forms the base of our solid and restricts the triple integral's \(xy\)-bounds.

Triple Integral

Triple integrals allow us to compute volumes and other properties of solids in three dimensions. The integral \(M_{xy} = \int_{-2}^{2} \int_{-(1/2)\sqrt{4-x^2}}^{(1/2)\sqrt{4-x^2}} \int_{0}^{2-x} z \, dz \, dy \, dx\) is used to determine the mass moment about the \(xy\)-plane.
In solving this, the process involves:

• Evaluating the innermost integral over \(z\), the vertical component, ranging from \(0\) to \(2-x\).
• The intermediate integral over \(y\), the width, ranging from \(-(1/2)\sqrt{4-x^2}\) to \((1/2)\sqrt{4-x^2}\).
• The outermost integral over \(x\), from \(-2\) to \(2\), spans the overall length of the cylinder.

Each integral captures a spatial dimension of the solid, and together, they form a complete view of the geometry involved. The result constitutes the value of \(M_{xy}\), providing insight into how mass is oriented relative to the horizontal plane.

Mass Moment

The mass moment, sometimes referred to as the moment of mass, provides a measure of an object's distribution of mass concerning a particular plane or axis. In the context of this exercise, we are calculating \(M_{xy}\), the mass moment about the \(xy\)-plane.
The significance of this measurement lies in understanding the balance and center of mass of the solid. By computing this value, one can deduce how the material's weight is spatially distributed.
In mathematical terms, the mass moment can be imagined as the 'weighted average', where each element of the mass contributes based on its position along the axis of interest. Here, the value of \(M_{xy}\) is used in verifying \(\overline{z}\) location in the ensuing steps of solid analysis.

Symmetry Analysis

Symmetry is a powerful property when solving complex mathematical problems, as it often simplifies calculations by revealing balance points. In this exercise, symmetry analysis shows that the solid's center of mass lies on the \(z\)-axis.
The elliptical cylinder and its bounding are symmetric about the \(x\)-axis and the \(y\)-axis. This symmetry means that any deviations in mass distribution cancel out horizontally, informing us that:

• \(\overline{x} = 0\)
• \(\overline{y} = 0\)

Thus, these points don’t need detailed computation because the balanced structure leads directly to these conclusions. Symmetry lets us assert the position of the center of mass more efficiently than integrating for explicit \(x\) and \(y\) coordinates, which prove unequivocally aligned with the axes in symmetric conditions.