To find the area bounded by the curves $y = x^2 + 2$ and $y = 3x^2$, we follow these steps:

Step 1: Find the points of intersection

Set the two equations equal to each other to find the $x$-coordinates where the curves intersect.

$x^2 + 2 = 3x^2$

Subtract $x^2$ from both sides:

$2 = 2x^2$

Now divide by 2:

$1 = x^2$

Taking the square root of both sides:

$x = \pm 1$

Thus, the curves intersect at $x = -1$ and $x = 1$.

Step 2: Set up the integral for the area

The area between the curves is given by the integral of the difference between the functions $y = 3x^2$ and $y = x^2 + 2$ over the interval from $x = -1$ to $x = 1$.

The area $A$ is:

$A = \int_{-1}^{1} \left( (3x^2) - (x^2 + 2) \right) \, dx$

Simplify the integrand:

$A = \int_{-1}^{1} \left( 3x^2 - x^2 - 2 \right) \, dx$

$A = \int_{-1}^{1} \left( 2x^2 - 2 \right) \, dx$

Step 3: Compute the integral

Now, integrate the function:

$A = \int_{-1}^{1} 2(x^2 - 1) \, dx$

First, compute the antiderivative of $2(x^2 - 1)$:

$\int 2(x^2 - 1) \, dx = 2\left( \frac{x^3}{3} - x \right)$

Now, evaluate the integral from $-1$ to $1$:

$A = \left[ 2\left( \frac{x^3}{3} - x \right) \right]_{-1}^{1}$

Substitute the upper and lower limits:

$A = 2\left( \left( \frac{1^3}{3} - 1 \right) - \left( \frac{(-1)^3}{3} - (-1) \right) \right)$

$A = 2\left( \left( \frac{1}{3} - 1 \right) - \left( \frac{-1}{3} + 1 \right) \right)$

$A = 2\left( \left( \frac{1}{3} - \frac{3}{3} \right) - \left( \frac{-1}{3} + \frac{3}{3} \right) \right)$

$A = 2\left( \left( -\frac{2}{3} \right) - \left( \frac{2}{3} \right) \right)$

$A = 2\left( -\frac{4}{3} \right)$

$A = -\frac{8}{3}$

Since area cannot be negative, we take the absolute value:

$A = \frac{8}{3}$

Thus, the area bounded by the curves is $\boxed{\frac{8}{3}}$.

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Further Questions:

1. How do you calculate the area between curves for other types of functions?
2. What is the general method for finding the area between two curves using integrals?
3. How would the area change if the curves had different forms (e.g., linear vs. quadratic)?
4. Can you explain the significance of symmetry in this problem?
5. How do you handle areas that are above and below the x-axis when solving similar problems?

Tip:

When working with definite integrals to find areas, always ensure that you subtract the lower function from the upper function, particularly when the curves intersect at more than one point.