Question

Find the centroid of the region bounded by the given curves: y = x³, xy = 30, y = 0. The coordinates of the centroid are (x, y) =

Answer 1

To find the centroid of the region bounded by the curves y = x³, xy = 30, and y = 0, follow these steps: find the area of the region by integrating the curves, calculate the x-coordinate of the centroid, and then find the y-coordinate of the centroid.

Step-by-step explanation:

To find the centroid of the region bounded by the curves y = x³, xy = 30, and y = 0, we need to determine the x-coordinate and y-coordinate of the centroid.

To start, let's find the area of the region by integrating the curves.

1. For the curve y = x³, we integrate from 0 to c, where c is the x-coordinate of the point where y = 0 (the intersection point of y = x³ and y = 0)
2. For the curve xy = 30, we solve for y in terms of x: y = 30/x
3. We integrate y = 30/x from c to d, where d is the x-coordinate of the point where y = 0 (the intersection point of y = 0 and xy = 30)
4. We can now calculate the area of the region using the definite integral formula: A = ∫(from c to d) (y₂ - y₁) dx

Once we have the area, we can find the x-coordinate of the centroid using the formula: x = (∫(from c to d) (x * (y₂ - y₁)) dx) / A

Finally, the y-coordinate of the centroid can be found by evaluating the definite integral: y = (∫(from c to d) (0.5 * (y₂² - y₁²)) dx) / A

So, the coordinates of the centroid are (x, y).