Question

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

Answer 1

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).