Centroids of Planar Regions Using Integral Calculus

The centroid of a planar region is the point where the area can be considered to be concentrated, representing the geometric center of the shape. Using integral calculus, the centroid coordinates $(\overline{x}, \overline{y})$ are calculated as weighted averages of the coordinates, weighted by the differential area elements and divided by the total area $A$. The area of the region is given by $\displaystyle A = \int dA$, with represented as for horizontal strips or for vertical strips. For regions bounded by curves (upper) and (lower) over , the area is . The -coordinate of the centroid is given by . Symmetry in the figure simplifies calculations, as the centroid lies on the axis of symmetry, making the coordinate along that axis equal to the symmetry line's coordinate. Centroid calculations are crucial in engineering for structural analysis, material distribution, moments of inertia computation, and design of balanced mechanical systems. Integration enables precise determination of centroids for irregular shapes where simple formulas are not applicable.