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

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

### 1. Find the points of intersection of the two curves
Set \(6x^2 = x^2 + 6\) to find the intersection points:

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

Thus, the points of intersection are \(x = -\frac{\sqrt{30}}{5}\) and \(x = \frac{\sqrt{30}}{5}\).

### 2. Compute the area of the region
The area \(A\) of the region is found by integrating the difference of the two curves over the interval \([- \frac{\sqrt{30}}{5}, \frac{\sqrt{30}}{5}]\):

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

Now, calculate the area by splitting the integral:

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

We will compute these integrals step by step.

### 3. Compute the centroid coordinates
The \(x\)-coordinate \(\bar{x}\) of the centroid is given by:

\[
\bar{x} = \frac{1}{A} \int_{-\frac{\sqrt{30}}{5}}^{\frac{\sqrt{30}}{5}} x \left( (x^2 + 6) - 6x^2 \right) dx
\]

The \(y\)-coordinate \(\bar{y}\) of the centroid is given by:

\[
\bar{y} = \frac{1}{A} \int_{-\frac{\sqrt{30}}{5}}^{\frac{\sqrt{30}}{5}} \frac{1}{2} \left( (x^2 + 6)^2 - (6x^2)^2 \right) dx
\]

Let me calculate each integral for you!The centroid of the region bounded by the curves \(y = 6x^2\) and \(y = x^2 + 6\) is:

- \( \bar{x} = 0 \)
- \( \bar{y} = \frac{11}{10} \)

Thus, the centroid is located at the point \(\left(0, \frac{11}{10}\right)\).

Do you need further clarification or more details?

Here are some related questions:
1. How does symmetry affect the x-coordinate of the centroid?
2. What would happen to the centroid if one of the curves changed?
3. Can this method be generalized to other parabolic curves?
4. How does the area of the region influence the location of the centroid?
5. What are the applications of centroids in physics and engineering?

**Tip:** When curves are symmetric with respect to the y-axis, the x-coordinate of the centroid will often be zero.

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

Learn how to calculate the centroid of the region bounded by the curves y = 6x^2 and y = x^2 + 6. This detailed step-by-step guide uses integral calculus, includes symmetry considerations, and applies key concepts in calculus. Suitable for undergraduate calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation