Final answer:
The region is bounded by the parabola y = 64 - x² and the x-axis. The region's centroid can be visually estimated and calculated exactly using integral calculus, with x-bar being 0 due to symmetry and y-bar being computed using the area and moment about the x-axis.
Step-by-step explanation:
The question asks to sketch the region bounded by the curves y = 64 - x² and y = 0, estimate the location of the centroid visually, and then find the exact coordinates of the centroid.
To sketch the region, we first graph the equation y = 64 - x². This is a downward-opening parabola with vertex at (0, 64). The curve intersects the x-axis at points where y = 0, which occurs at x = -8 and x = 8.
Next, we find the exact coordinates of the centroid (x-bar, y-bar) where x-bar is the average x-value and y-bar is the average y-value of the region. To find x-bar, we use the formula:
x-bar = (1/Area) ∫ (x∙dA)
Since the region is symmetric about the y-axis, x-bar = 0. To find y-bar, we use the formula:
y-bar = (1/Area) ∫ (y∙dA)
Since the area between the curve y = 64 - x² and the x-axis from x = -8 to x = 8 is required for the calculation, we set up the integral:
Area = ∫_{-8}^{8} (64 - x²) dx
Through integration we find the area A and then calculate:
y-bar = (1/A) ∫_{-8}^{8} x∙(64 - x²) dx
This computation will provide us with the exact y-coordinate of the centroid.