Problem 26 - 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. The solid generated by revolving the region bounded by \(y=x^{2}, x=2,\) and the \(x\) -axis about the \(x\) -axis.

Step-by-Step Solution

Answer

The centroid coordinates are \( \left( \frac{5}{3}, 0, 0 \right) \).

Step 1: Understand the shape

The problem describes a figure obtained by revolving the region under the curve of \( y = x^2 \) between \( x = 0 \) and \( x = 2 \) about the \( x \)-axis. This creates a 3D solid known as a paraboloid.

Step 2: Set up the integral

To find the centroid of the solid, use the formula for the centroid \( \bar{x}, \bar{y}, \bar{z} \) of a volume of revolution. When revolving around the \( x \)-axis, \( \bar{y} = \bar{z} = 0 \) due to symmetry, so we only need to find \( \bar{x} \) using\[\bar{x} = \frac{1}{V} \int_{a}^{b} x A(x) \, dx\]where \( A(x) \) is the area of the cross-section at \( x \), which is \( \pi (x^2)^2 = \pi x^4 \), and \( V \) is the total volume.

Step 3: Find the total volume

The total volume is calculated by the integral\[V = \int_{0}^{2} \pi x^4 \, dx\]Evaluate the integral:\[V = \pi \left[ \frac{x^5}{5} \right]_{0}^{2} = \pi \left( \frac{32}{5} \right) = \frac{32\pi}{5}\]

Step 4: Calculate \( \bar{x} \)

Now substitute \( A(x) = \pi x^4 \) into the centroid formula:\[\bar{x} = \frac{1}{V} \int_{0}^{2} x(\pi x^4) \, dx = \frac{1}{\frac{32\pi}{5}} \int_{0}^{2} \pi x^5 \, dx\]\[= \frac{5}{32\pi} \cdot \pi \left[ \frac{x^6}{6} \right]_{0}^{2} = \frac{5}{32} \cdot \frac{64}{6} = \frac{10}{6} = \frac{5}{3}\]

Step 5: State the coordinates of the centroid

The centroid of the volume is located at \( \left( \frac{5}{3}, 0, 0 \right) \) using symmetry and the calculated \( \bar{x} \).

Key Concepts

Volume of Revolution

When we talk about the volume of revolution, we are essentially discussing the 3D shape formed when a 2D region is revolved around a line. This process turns a flat surface into a solid object. In this problem, the region under the curve \( y = x^2 \) between \( x = 0 \) and \( x = 2 \) is revolved around the \( x \)-axis.

• This action creates a shape known as a paraboloid, which is very similar to the shape of a bowl or a wine glass when viewed from the side.
• To correctly find the volume of this revolved shape, we use integral calculus. The volume is determined by slicing the solid into many thin disks and adding up their volumes.
• The cross-sectional area, \( A(x) \), of each disk as a function of \( x \) is \( \pi x^4 \), derived from rotating \( (x^2)^2 \) around the axis.

In this exercise, by integrating over the specified range, we can find the total volume of the solid.

Integral Calculus

Integral calculus is a fundamental tool used to calculate areas under curves, volumes of solids, and other related concepts in mathematics. In this particular problem, we use it to find the total volume of a paraboloid and its centroid.

• The fundamental idea is to break down a complex shape into smaller and more manageable pieces, such as slices or disks.
• By using integration, we sum these pieces to find the overall volume or other quantities of interest.
• In our paraboloid example, the function \( y = x^2 \) creates slices of the shape perpendicular to the \( x \)-axis, each represented by \( \pi x^4 \).

We then integrate this function from \( 0 \) to \( 2 \) to find the complete volume, which is \( \frac{32\pi}{5} \). Integral calculus allows us to handle these calculations and is crucial in finding centroids of complex shapes.

Paraboloid

A paraboloid is a three-dimensional shape that is created by revolving a parabolic curve around an axis.

• In our specific case, the parabola \( y = x^2 \) created between \( x = 0 \) and \( x = 2 \) is revolved around the \( x \)-axis.
• This transformation results in what is known as a paraboloid of revolution, whose surface is shaped somewhat like an inverted cone but with a curved surface instead of a sharp edge.
• The paraboloid's surface increases quadratically as you move outward from the axis of rotation, which is key to understanding its volume.

The calculation of centroids—the weighted average positions of all points in a body—requires recognizing the symmetry and shape of the paraboloid, allowing simpler computations for coordinates in some cases. In this problem, such symmetry gives us \( \bar{y} = \bar{z} = 0 \), leading to a simpler focus on finding \( \bar{x} \).