How do you find the centroid of the region bounded by the curves y = x^3, y = 2, and y = 0?

To find the centroid of the region bounded by the curves given, we need to follow a few steps.

First, we need to visualize the area we are dealing with. The curves mentioned are:

• y = x³
• y = 2
• y = 0 (the x-axis)

Next, we need to determine the points of intersection of these curves to know the limits of integration. Setting y = x³ equal to y = 2, we have:

x³ = 2

which gives us:

x = 2^1/3 or approximately x = 1.26.

The next intersection is with the x-axis. Setting y = 0 gives us x = 0 since y = x³ at x = 0 is also 0.

Now we have the bounds of integration as x = 0 to x = 2^1/3.

We can find the area A of the region as follows:

A = ∫₀^21/3 (2 – x³) dx.

Calculating this integral:

A = 2x – (1/4)x⁴ | from 0 to 2^1/3

Plugging in the bounds:

A = [2(2^1/3) – (1/4)(2^1/3)⁴] – [0] = 2(2^1/3) – (1/4)(4) = 2(2^1/3) – 1.

Now we compute the coordinates of the centroid (x̄, ȳ) using the formulas:

x̄ = (1/A) * ∫₀^21/3 x(2 – x³) dx

and

ȳ = (1/(2A)) * ∫₀^21/3 (2 – x³)² dx.

This process will yield the coordinates of the centroid. Solving these integrals will require some algebra, and you might find it beneficial to use a calculator for the numerical parts.

In conclusion, the centroid represents the center of mass of the area bounded by the curves and can be represented as the point (x̄, ȳ) after completing the integration and simplifications.