Problem 6 - Thomas Calculus

Question

The solids in Exercises \(1-12\) all have constant density \(\delta=1\) 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

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

Step 1: Understanding the Problem

We need to find the center of mass for a solid with uniform density, given the boundaries: the plane \(z=0\), the elliptical cylinder \(x^2 + 4y^2 = 4\), and the plane \(z = 2 - x\). The goal is to compute \(\overline{x}\), \(\overline{y}\), and verify \(\overline{z} = \frac{5}{4}\).

Step 2: Recognizing Symmetry for X and Y Coordinates

Due to symmetry of the elliptical cylinder equation \(x^2 + 4y^2 = 4\), the center of mass in the \(x-y\) plane lies at the origin. Thus, \(\overline{x} = 0\) and \(\overline{y} = 0\).

Step 3: Setting Up the Triple Integral for \(M\)

The mass \(M\) of the solid can be calculated using the triple integral:\[M = \int_{-2}^{2} \int_{-(1/2) \sqrt{4-x^2}}^{(1/2) \sqrt{4-x^2}} \int_{0}^{2-x} 1 \; dz \; dy \; dx\]

Step 4: Computing the Inner Integral

Integrate the innermost integral with respect to \(z\):\[\int_{0}^{2-x} 1 \; dz = [z]_{0}^{2-x} = 2-x\]

Step 5: Evaluating the Second Integral

Substitute the result of the previous integration and evaluate:\[\int_{-(1/2) \sqrt{4-x^2}}^{(1/2) \sqrt{4-x^2}} (2-x) \; dy = (2-x) \cdot \sqrt{4-x^2}\]

Step 6: Final Integration and Finding Mass

Integrate with respect to \(x\):\[M = \int_{-2}^{2} (2-x) \cdot \sqrt{4-x^2} \; dx\]This integral requires an integral table or a numerical approach. Calculate this to find the mass.

Step 7: Evaluating Moment \(M_{xy}\)

Set up the integral for \(M_{xy}\) which incorporates an additional factor of \(z\):\[M_{xy} = \int_{-2}^{2} \int_{-(1 / 2) \sqrt{4-x^{2}}}^{(1 / 2) \sqrt{4-x^{2}}} \int_{0}^{2-x} z \; dz \; dy \; dx\]

Step 8: Computing Inner Integral for \(M_{xy}\)

Integrate with respect to \(z\):\[\int_{0}^{2-x} z \; dz = \left[\frac{z^2}{2}\right]_{0}^{2-x} = \frac{(2-x)^2}{2}\]

Step 9: Evaluating Intermediate Integrals for \(M_{xy}\)

Evaluate the integral with respect to \(y\):\[\int_{-(1/2)\sqrt{4-x^2}}^{(1/2)\sqrt{4-x^2}} \frac{(2-x)^2}{2} \; dy = \frac{(2-x)^2}{2} \cdot \sqrt{4-x^2}\]

Step 10: Final Integral for \(M_{xy}\)

Integrate with respect to \(x\):\[M_{xy} = \int_{-2}^{2} \frac{(2-x)^2}{2} \cdot \sqrt{4-x^2} \; dx\]Use integral tables to evaluate this expression.

Step 11: Verifying \(\overline{z}\)

Finally, divide \(M_{xy}\) by \(M\) to confirm \(\overline{z} = \frac{5}{4}\).

Key Concepts

Elliptical Cylinder

An elliptical cylinder is a three-dimensional shape that is quite similar to a regular circular cylinder, but its cross-section is an ellipse rather than a circle. This means that the sides of the cylinder are bounded by an ellipse, defined by an equation such as \(x^2 + 4y^2 = 4\), which was used in the problem here. The ellipse defines the boundaries of the solid around the axis, creating a continuous but oval-like cage.

By understanding how an elliptical cylinder operates in geometry, one can visualize its symmetry properties. A critical aspect is that this symmetric nature implies that certain calculations, such as finding the center of mass in this shape, can be simplified under the right conditions. This is because the shape has a uniform distribution around the x and y axes, helping to determine that \(\overline{x} = 0\) and \(\overline{y} = 0\) based on symmetry.

Triple Integral

A triple integral extends the concept of integration to three dimensions, enabling the calculation of volumes or other property distributions in 3D spaces. In this exercise, the triple integral is essential for determining the mass \(M\) and moment \(M_{xy}\) of the solid given by the elliptical cylinder and restricted by the planes.

The complete representation of a triple integral in this scenario is:

• \(\int_{-2}^{2} \int_{-(1/2)\sqrt{4-x^2}}^{(1/2)\sqrt{4-x^2}} \int_{0}^{2-x} f(x,y,z) \; dz \; dy \; dx\)

A triple integral contains three levels of integration: with respect to \(z\), then \(y\), and finally \(x\). Here, we used these integrals to add up "slices" of volume or mass across these dimensions. The problem has been solved using these layers of integration to compute first volumes and then specialized measurements like centers of mass or other spatial properties.

Constant Density

The assumption of constant density simplifies many calculations in physics and engineering. In this problem, the solid is given a uniform density of \(\delta = 1\), which means that the density doesn't change at any point within the solid's volume.

Having a constant density translates to the fact that mass is only a function of volume. Using a density of \(1\) eliminates any extra complexities in solving for mass or center of mass by making calculations direct and purely volumetric. This uniformity in density feeds into simplifications during both integration and symmetry arguments in the calculation steps.

Symmetry in Geometry

Symmetry plays a crucial role in geometrical problem-solving, often allowing complicated problems to become much simpler. The elliptical cylinder \(x^2 + 4y^2 = 4\) is symmetric about both the x and y axes. This symmetry means that any moment from mass on one side of the origin is balanced by an equal yet opposite moment from mass on the other side.

In computing the center of mass, the symmetry simplifies the calculation by indicating that \(\overline{x} = 0\) and \(\overline{y} = 0\) as mass centers on these axes. Knowing about symmetrical properties can drastically cut down on unnecessary calculations while ensuring that the results remain accurate and clear, especially in three-dimensional math scenarios such as this one.