Find the centroid of the region bounded by the graphs of the functions  y = 6x^2, y = x^2 + 6

To find the centroid of the region bounded by the curves \( y = 6x^2 \) and \( y = x^2 + 6 \), we need to follow these steps:

### Step 1: Find the points of intersection of the curves
We set the two equations equal to each other to determine the points where the curves intersect:

\[
6x^2 = x^2 + 6
\]

Subtract \( x^2 \) from both sides:

\[
5x^2 = 6
\]

Solve for \( x \):

\[
x^2 = \frac{6}{5}, \quad x = \pm \sqrt{\frac{6}{5}} = \pm \frac{\sqrt{30}}{5}
\]

Thus, the curves intersect at \( x = \pm \frac{\sqrt{30}}{5} \).

### Step 2: Calculate the area of the region
The area of the region is found by integrating the difference between the two curves over the interval from \( x = -\frac{\sqrt{30}}{5} \) to \( x = \frac{\sqrt{30}}{5} \). The area \( A \) is given by:

\[
A = \int_{-\frac{\sqrt{30}}{5}}^{\frac{\sqrt{30}}{5}} \left( (x^2 + 6) - 6x^2 \right) \, dx = \int_{-\frac{\sqrt{30}}{5}}^{\frac{\sqrt{30}}{5}} \left( -5x^2 + 6 \right) \, dx
\]

We can break this into two integrals:

\[
A = \int_{-\frac{\sqrt{30}}{5}}^{\frac{\sqrt{30}}{5}} 6 \, dx - \int_{-\frac{\sqrt{30}}{5}}^{\frac{\sqrt{30}}{5}} 5x^2 \, dx
\]

Calculate each integral:

1. For the first term:

\[
\int_{-\frac{\sqrt{30}}{5}}^{\frac{\sqrt{30}}{5}} 6 \, dx = 6 \left( \frac{\sqrt{30}}{5} - \left(-\frac{\sqrt{30}}{5}\right) \right) = 6 \times \frac{2\sqrt{30}}{5} = \frac{12\sqrt{30}}{5}
\]

2. For the second term:

\[
\int_{-\frac{\sqrt{30}}{5}}^{\frac{\sqrt{30}}{5}} 5x^2 \, dx = 5 \times \frac{2}{3} \left( \frac{\sqrt{30}}{5} \right)^3 = \frac{5 \times 2 \times 30\sqrt{30}}{15} = \text{(Expression after expansion)}
\]



Find the centroid of the region bounded by the graphs of the functions y = 6x^2, y = x^2 + 6

This problem demonstrates how to find the centroid of a region bounded by two parabolic curves: y = 6x^2 and y = x^2 + 6. It covers key calculus concepts, including finding intersection points, calculating areas between curves, and determining the centroid using integration. Ideal for university-level calculus students.

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