locate the centroid x of the shaded area figure 1
Introduction
When studying planar figures in calculus or engineering, one of the most common tasks is to locate the centroid x of the shaded area figure 1. The centroid represents the geometric center of a shape, and determining its horizontal coordinate (often denoted as x̄) is essential for analyzing moments, forces, and stability. This article walks you through a clear, step‑by‑step process to find the centroid’s x‑coordinate, explains the underlying scientific principles, and answers frequently asked questions. By the end, you will have a solid methodological toolkit that can be applied to a wide range of similar problems.
Understanding the Geometry
Shape Description
Figure 1 typically depicts a composite region bounded by simple curves such as straight lines, parabolas, or circles. The shaded portion may be irregular, but its boundaries are always defined mathematically. For the purpose of this guide, assume the figure consists of:
- A base rectangle of width b and height h - A semicircular cut‑out of radius r positioned on the top edge
- A triangular extension attached to the right side The exact dimensions are given in the problem statement, but the methodology remains the same regardless of the specific numbers.
Identifying Sub‑areas Because the overall shape is often composite, it is helpful to divide the shaded area into simpler sub‑areas (e.g., rectangles, triangles, semicircles). Each sub‑area has its own centroid, and the overall centroid can be obtained by weighting these individual centroids.
Methodology for Finding the Centroid x The centroid’s x‑coordinate for a composite area is calculated using the formula:
[ \bar{x} = \frac{\sum A_i \bar{x}_i}{\sum A_i} ]
where:
- (A_i) is the area of the i‑th sub‑shape
- (\bar{x}_i) is the x‑coordinate of that sub‑shape’s centroid
This equation essentially computes a weighted average of the individual centroids, weighted by their respective areas.
Steps to Apply the Formula
- Break the figure into basic shapes – Identify each primitive shape that composes the shaded region.
- Calculate the area of each shape – Use standard geometric formulas (e.g., (A_{\text{rect}} = \text{length} \times \text{width})). 3. Determine the centroid x of each shape – For simple shapes, the centroid formulas are well‑known:
- Rectangle: (\bar{x} = \frac{\text{left} + \text{right}}{2})
- Triangle: (\bar{x} = \frac{\text{base}}{3}) from the base vertex
- Semicircle: (\bar{x} = 0) if symmetric about the y‑axis, otherwise derived from integration.
- Compute the moment about the y‑axis – Multiply each area by its centroid x: (A_i \bar{x}_i).
- Sum the moments and total area – Add all (A_i \bar{x}_i) values and all (A_i) values separately.
- Divide total moment by total area – The quotient yields the overall (\bar{x}).
Step‑by‑Step Calculation
Below is a concrete example that illustrates each step. Suppose the dimensions are:
- Rectangle: width = 10 units, height = 4 units - Semicircle (cut‑out): radius = 2 units, centered at x = 5 units
- Triangle: base = 6 units, height = 3 units, attached at x = 8 units
1. Compute Individual Areas
- (A_{\text{rect}} = 10 \times 4 = 40)
- (A_{\text{semi}} = \frac{1}{2} \pi r^2 = \frac{1}{2} \pi (2)^2 = 2\pi) (subtracted later)
- (A_{\text{tri}} = \frac{1}{2} \times 6 \times 3 = 9)
2. Find Each Shape’s Centroid x
- Rectangle: (\bar{x}_{\text{rect}} = \frac{0 + 10}{2} = 5)
- Semicircle (cut‑out): Because it is symmetric about x = 5, (\bar{x}_{\text{semi}} = 5)
- Triangle: The centroid of a triangle measured from the base is at (\frac{1}{3}) of its height. If the base lies on the line x = 8, then (\bar{x}_{\text{tri}} = 8 + \frac{6}{3} = 10)
3. Adjust for the Cut‑Out
Since the semicircle is removed, its contribution is subtracted:
- Adjusted moment for the semicircle: (-A_{\text{semi}} \times \bar{x}_{\text{semi}} = -2\pi \times 5)
4. Sum Moments
[ \text{Total moment} = (40 \times 5) + (-2\pi \times 5) + (9 \times 10) ]
[\text{Total moment} = 200 - 10\pi + 90 = 290 - 10\pi]
5. Sum Areas
[ \text{Total area} = 40 - 2\pi + 9 = 49 - 2\pi ]
6. Compute (\bar{x})
[\bar{x} = \frac{290 - 10\pi}{49 - 2\pi} ]
Evaluating numerically (using (\pi \approx 3.1416)):
[ \bar{x} \approx \frac{290 - 31.416}{49 - 6.283} = \frac{
