Final answer:
To find the centroid of the planar region bounded by y = 4 - x² and y = x + 2, we need to find the points of intersection between the two curves and then calculate the average of the y-values of these points. The centroid of the planar region is (1, 1.5).
Step-by-step explanation:
To find the centroid of the planar region bounded by y = 4 - x² and y = x + 2, we need to first find the points of intersection between the two curves. Setting the equations equal to each other, we have:
4 - x² = x + 2
Combining like terms and rearranging, we get:
x² + x - 2 = 0
Using the quadratic formula, we find that x = -2 and x = 1. Therefore, the points of intersection are (-2, 0) and (1, 3).
To find the y-coordinate of the centroid, we need to find the average of the y-values of the points of intersection. Adding the y-values together and dividing by 2, we get:
(0 + 3)/2 = 1.5
So, the centroid of the planar region is (1, 1.5).