Final answer:
The centroid of an area bounded by curves is calculated by estimating the area under the curves with geometric shapes such as triangles and rectangles, finding their individual centroids and areas, and then computing the weighted average position of these shapes' centroids.
Step-by-step explanation:
Calculating the Centroid of an Area Bounded by Curves
To calculate the centroid of an area bounded by curves, we must first understand the concept of the centroid itself. The centroid is essentially the center of mass of a shape, assuming the shape has uniform density. It is the point where the shape would balance perfectly if it were to be placed on a pin. The process of finding a centroid involves integrating the geometry's shape in terms of its area.
One way to estimate the area under a curve for the purpose of finding the centroid is by simplifying the area into geometric shapes whose centroids we can easily determine. For example, we could approximate an irregular area under the curve by breaking it down into a series of triangles and rectangles. A method mentioned is approximating the area under a curve by using a right triangle, as in Figure 10.13, or drawing a rectangle around it, as in Example 13.2.3.
After determining the areas and centroids of these simplified shapes, we use the principle of moments to find the overall centroid by taking the weighted average of the centroids of the individual shapes, with the weights being the areas of these shapes. For irregular shapes, this would be the integral of the product of the area density function and the distance from a reference axis, typically the x or y axis, over the region of interest.