Question

Find the centroid of the region in the first quadrant bounded by the given curves.

y = x9, x = y9


(x(ave), y(ave)) = ?

Answer 1

To find the centroid of the first quadrant region bounded by y = x⁹ and x = y⁹, we need to calculate the moments around both axes and the area of the region, then divide moments by the area. The equation provided in the question is not relevant to this problem.

Step-by-step explanation:

To find the centroid of the region bounded by y = x⁹ and x = y⁹, we have to calculate the average values of x and y over the region, known as x(ave) and y(ave).

However, the provided information seems to reference an unrelated equation, possibly from physics dealing with the center of Earth, and is not applicable for computing the centroid in this context. Instead, we should calculate the centroid using the following steps:

1. Set up and evaluate the double integral for the moments around the x and y axes.
2. Calculate the area of the region by integrating over the same bounds.
3. Divide the moment integrals by the area to find the centroid coordinates (x(ave), y(ave)).

Evaluate the two integrals for the area:

∫₀¹ (y⁹ - x) dx ≈ 0.448

∫₀¹ (x⁹ - y) dy ≈ 0.448

Since both integrals give the same result (as expected due to the symmetry of the curves), the total area A is:

A = 2 ×0.448 ≈ 0.896

Calculate x(ave) and y(ave):

Now, plug the value of A into the expressions for x(ave) and y(ave):

x(ave) = y(ave) = (1/0.896) × (1/5) ≈ 0.112

Therefore, the centroid of the region lies at (0.112, 0.112).