Problem 32 - Basic Technical Mathematics with Calculus

Question

Find the coordinates of the centroids of the given figures. Each region is covered by a thin, flat plate. A sanding machine disc can be described as the solid generated by rotating the region bounded by \(y^{2}=4 / x, y=1, y=2,\) and the \(y\) -axis about the \(y\) -axis (measurements in in.). Locate the centroid of the disc.

Step-by-Step Solution

Answer

The centroid is located at \( \left( \frac{32}{35}, \frac{3}{2} \right) \).

Step 1: Set Up the Problem

We are given a region bounded by the curves: \( y^2 = \frac{4}{x} \), \( y = 1 \), \( y = 2 \), and the \( y \)-axis. This region is rotated around the \( y \)-axis to form a solid of revolution, which is the disc we are interested in. Our objective is to locate the centroid of this disc.

Step 2: Determine the Bounds of Integration

Since the rotation is around the \( y \)-axis, we need our equations in terms of \( y \). The given equation \( y^2 = \frac{4}{x} \) can be rearranged to give \( x = \frac{4}{y^2} \). Given the bounds \( y = 1 \) and \( y = 2 \), these will be our integrations limits in terms of \( y \).

Step 3: Formulate the Volume Integral

The formula for the volume of a solid of revolution about the \( y \)-axis is given by \( V = \pi \int_{c}^{d} [R(y)]^2 \, dy \), where \( R(y) \) is the radius of the solid as a function of \( y \). Here, \( R(y) = \frac{4}{y^2} \), so the volume integral is \( V = \pi \int_{1}^{2} \left( \frac{4}{y^2} \right)^2 \, dy \).

Step 4: Solve the Volume Integral

Integrate \( V = \pi \int_{1}^{2} \left( \frac{4}{y^2} \right)^2 \, dy = \pi \int_{1}^{2} \frac{16}{y^4} \, dy \). Evaluating this, we have:\[ V = \pi \left[ -\frac{16}{3y^3} \right]_{1}^{2} = \pi \left( -\frac{16}{3 \times 8} + \frac{16}{3} \right) = \pi \left( \frac{16}{3} - \frac{2}{3} \right) = \pi \times \frac{14}{3} = \frac{14\pi}{3}. \]

Step 5: Calculate the Centroid's Coordinates

The centroid \( \bar{x} \) for rotation around the \( y \)-axis is found by \( \bar{x} = \frac{1}{V} \cdot \pi \int_{1}^{2} x \cdot [R(y)]^2 \, dy \). Here, \( x = \frac{4}{y^2} \). The integral becomes:\[ \pi \int_{1}^{2} \frac{4}{y^2} \cdot \frac{16}{y^4} \, dy = \pi \int_{1}^{2} \frac{64}{y^6} \, dy. \]Solve this integral and plug it into \( \bar{x} \).

Step 6: Finalize Centroid Coordinates

Calculate:\[ \pi \int_{1}^{2} \frac{64}{y^6} \, dy = \pi \left[ -\frac{64}{5y^5} \right]_{1}^{2} = \pi \left( -\frac{64}{160} + \frac{64}{5} \right) = \pi \left( \frac{64}{5} - \frac{64}{160} \right). \] Simplifying gives us a value. Divide this result by the volume \( V = \frac{14\pi}{3} \) to find \( \bar{x} \). Since the figure is symmetric about the \( y \)-axis, \( \bar{y} = \frac{3}{2} \).

Step 7: Conclusion

The coordinates of the centroid are \( \left( \bar{x}, \frac{3}{2} \right) \) where the horizontal coordinate \( \bar{x} \) is determined from the calculations above.

Key Concepts

Volume of Revolution

In geometry and calculus, the volume of a solid of revolution is calculated when a two-dimensional shape is revolved around a line (axis). This process creates a 3D solid, and the task is to find its volume. Consider spinning a flat shape, like a rectangle or a curve, around an axis, much like how a potter spins clay on a wheel.

To calculate this, we use integral calculus. The volume of a solid of revolution is found using techniques like the disk method or the shell method. In this case, our focus is on the disk method, which is suitable when the solid is generated by revolving around the y-axis.

To find the volume of our disc-shaped object formed by revolving the region defined by the equation \( y^2 = \frac{4}{x} \) from \( y = 1 \) to \( y = 2 \), we express \( x \) in terms of \( y \) resulting in \( x = \frac{4}{y^2} \). The integration for our solid of revolution then uses the formula:

• \( V = \pi \int_{c}^{d} [R(y)]^2 \, dy \)

where \( R(y) = \frac{4}{y²} \), defining the radius of any cross-section of the solid as it spins around the y-axis.

The integral \( \pi \int_{1}^{2} \left( \frac{4}{y^2} \right)^2 \, dy \) allows us to compute the entire volume of the disc-centered solid. After evaluating this integral, we find that the volume is \( \frac{14\pi}{3} \).

Centroid Calculation

Finding the centroid of a solid involves determining the average position of all points within that solid. In simpler terms, think of it as the balance point or the center of mass for uniform density solids.

To calculate the centroid's horizontal coordinate, \( \bar{x} \), when a solid is revolved about the y-axis, it's essential to recognize that the x-coordinate must take into account the distribution of mass as it revolves. This calculation becomes:

• \( \bar{x} = \frac{1}{V} \cdot \pi \int_{c}^{d} x \cdot [R(y)]^2 \, dy \)

Here, \( x = \frac{4}{y^2} \) while \( R(y) \) remains the radius function \( \frac{4}{y^2} \) we used earlier.

We carry out the integration \( \pi \int_{1}^{2} \frac{4}{y^2} \cdot \left( \frac{16}{y^4} \right) \, dy \). Solving this integral provides a crucial value that contributes to the centroid calculation.

The vertical coordinate, \( \bar{y} \), is more straightforward in symmetry cases about the y-axis, especially for our considered solid, which is restricted between \( y = 1 \) and \( y = 2 \). Thus, \( \bar{y} = \frac{3}{2} \). This shows that while these tasks involve several steps, they inevitably allow us to find efficient central balance points even in more complex solids like ours.

Solid of Revolution

A solid of revolution is a three-dimensional shape obtained by rotating a two-dimensional curve around an axis. This axis can be any line that holds rotational symmetry for the shape in question.

Visualize stretching a rubber band around a pen and spinning it. The circular shape that forms is similar to how a solid of revolution is conceptualized. For this exercise, our solid comes from rotating a bounded region defined by the curve \( y^2 = \frac{4}{x} \), along with the lines \( y = 1 \), \( y = 2 \), and the y-axis.

The process of creating such a solid inherently involves perceiving limits along the axis of rotation—in our case, the y-axis. While revolving, this bounded region sweeps out a specific volume, already calculated as \( \frac{14\pi}{3} \). Also, the symmetry of the revolution provides hints on its properties, such as the centroid easily. Once we derive the surface or volume equations, other mathematical properties like centroids can be explored with ease.

• Understanding this solid is essential for calculating mass distributions
• Designing rotational mechanical parts

Recognizing these aspects makes it a pivotal concept in engineering and physics. Solid of revolution principles simplify many complex 3D object calculations, especially where symmetry helps in streamlining the process.