Final answer:
To find the centroid of the region bounded by the curves x=y^2 and xy=2, one must solve for the area and use integration to calculate the centroid's x and y coordinates.
Step-by-step explanation:
The question asks us to find the centroid of the region bounded by the curves x = y^2 and xy = 2. First, we need to determine the points of intersection for these two curves by simultaneously solving the equations. Substituting x from the first equation into the second gives us y^3 = 2, leading to an intersection point at y = √[3]{2}. Now, we anticipate another intersection point where y = -√[3]{2} because the graph of y^2 is symmetric about the x-axis.
We need to set up the integrals for centroid calculation, which involves finding the area A of the region and then using the formulas for x-bar (α) and y-bar (β), where:
- α = (1/A) ∫ x ⋅ dA
- β = (1/A) ∫ y ⋅ dA
The area element dA can be considered as a thin vertical strip of width dx and height y between the curves. The integral for the area (A) would then be the integral of (y) from -√[3]{2} to √[3]{2}. The centroid components α and β can be found by integrating x⋅y over this same range for α, and 0.5⋅y^2 for β.
To find the exact coordinates of the centroid, explicit computation of these integrals is required, which falls under the subject of Calculus.