Find the centroid of the region bounded by the curve 𝑦 = 𝑥 2 y=x 2 , the 𝑥 x-axis, and the lines 𝑥 = 0 x=0 and 𝑥 = 1 x=1.

The centroid of a region in the xy-plane is a point (x̄, ȳ) where x̄ is the average x-coordinate of the points in the region and ȳ is the average y-coordinate.

The x-coordinate of the centroid (x̄) is given by the formula:

x̄ = (1/A) ∫[from a to b] x*f(x) dx

The y-coordinate of the centroid (ȳ) is given by the formula:

ȳ = (1/2A) ∫[from a to b] [f(x)]^2 dx

Where A is the area under the curve f(x) from a to b, given by:

A = ∫[from a to b] f(x) dx

In this case, the curve is y = x^2, the x-axis is y = 0, and the lines are x = 0 and x = 1. So, we have:

A = ∫[from 0 to 1] x^2 dx = [x^3/3] (from 0 to 1) = 1/3

x̄ = (1/A) ∫[from 0 to 1] xx^2 dx = (1/(1/3)) ∫[from 0 to 1] x^3 dx = 3[x^4/4] (from 0 to 1) = 3/4

ȳ = (1/2A) ∫[from 0 to 1] (x^2)^2 dx = (1/(2*(1/3))) ∫[from 0 to 1] x^4 dx = 3/2*[x^5/5] (from 0 to 1) = 3/10

So, the centroid of the region is (3/4, 3/10).