Find the centroid of the region bounded by y = 3x^2, y = 0, x = 4

To determine the centroid, we use the equations:

x⁻ = 1/A (∫ (x dA))

y⁻ = 1/A (∫ (y dA))

First, we evaluate the value of A and dA as follows:

A = ∫dA

A = ∫ydx

A = ∫3x^2 dx

A = 3x^3 / 3 from 0 to 4

A = x^3 from 0 to 4

A = 64

We use the equations for the centroid,

x⁻ = 1/A (∫ (x dA))

x⁻ = 1/64 (∫ (x (3x^2 dx)))

x⁻ = 1/64 (∫ (3x^3 dx)

x⁻ = 1/64 (3 x^4 / 4) from 0 to 4

x⁻ = 1/64 (192) = 3

y⁻ = 1/A (∫ (y dA))

y⁻ = 1/64 (∫ (3x^2 (3x^2 dx)))

y⁻ = 1/64 (∫ (9x^4 dx)

y⁻ = 1/64 (9x^5 / 5) from 0 to 4

y⁻ = 1/64 (9216/5) = 144/5

The centroid of the curve is found at (3, 144/5).