Question

Find exact coordinates of the centroid of the region bounded by
y = x³ , y = 8 , x =0

Answer 1

To find the coordinates of the centroid of the region bounded by y = x³, y = 8, and x = 0, use the formula for the centroid of a plane region.

Step-by-step explanation:

To find the coordinates of the centroid of the region bounded by y = x³, y = 8, and x = 0, we can use the formula for the centroid of a plane region:

1. First, find the area of the region by integrating the difference between the two curves with respect to x, from x = 0 to x = c, where c is the x-coordinate where the two curves intersect.
2. Next, find the x-coordinate of the centroid using the formula: x = (1/Area) * ∫(x * (f(x) - g(x))) dx, where f(x) is the upper curve, g(x) is the lower curve, and the integral is taken from x = 0 to x = c.
3. Finally, find the y-coordinate of the centroid using the formula: y = (1/Area) * ∫(0.5 * (f(x)² - g(x)²)) dx, where f(x) and g(x) are the upper and lower curves respectively, and the integral is taken from x = 0 to x = c.

Using these steps, we can find the exact coordinates of the centroid.