To find the area of the region bounded by the two functions $f(x) = 3e^x - 1$ and $g(x) = x^2 - 1$ on the interval $[-1, 1]$, we follow these steps:

Step 1: Set up the integral

We need to find the points where $f(x) = g(x)$, which will help us identify the regions where one function is above the other.

So, we solve for the intersection points by setting: $3e^x - 1 = x^2 - 1$ This simplifies to: $3e^x = x^2$

This equation doesn't have a simple analytical solution, so we check the interval $[-1, 1]$ to see where the functions intersect by evaluating $f(x)$ and $g(x)$ at critical points.

Step 2: Evaluate at critical points

At $x = -1$: $f(-1) = 3e^{-1} - 1 \approx 3(0.3679) - 1 \approx 1.1037 - 1 = 0.1037$ $g(-1) = (-1)^2 - 1 = 0$

At $x = 1$: $f(1) = 3e^{1} - 1 \approx 3(2.718) - 1 = 8.154 - 1 = 7.154$ $g(1) = 1^2 - 1 = 0$

At $x = 0$: $f(0) = 3e^{0} - 1 = 3(1) - 1 = 2$ $g(0) = 0^2 - 1 = -1$

Based on these values, $f(x)$ is above $g(x)$ over the entire interval $[-1, 1]$.

Step 3: Set up the integral for the area

The area between two curves is given by the integral of the difference between the functions: $\text{Area} = \int_{-1}^{1} (f(x) - g(x)) \, dx$

Thus, we need to compute: $\text{Area} = \int_{-1}^{1} \left( (3e^x - 1) - (x^2 - 1) \right) dx$ Simplifying the integrand: $\text{Area} = \int_{-1}^{1} \left( 3e^x - x^2 \right) dx$

Step 4: Compute the integral

We now split the integral into two parts: $\text{Area} = \int_{-1}^{1} 3e^x \, dx - \int_{-1}^{1} x^2 \, dx$

1. The integral of $3e^x$: $\int_{-1}^{1} 3e^x \, dx = 3 \left[ e^x \right]_{-1}^{1} = 3(e^1 - e^{-1}) = 3(2.718 - 0.3679) = 3(2.3501) = 7.0503$

2. The integral of $x^2$: $\int_{-1}^{1} x^2 \, dx = \left[ \frac{x^3}{3} \right]_{-1}^{1} = \frac{1^3}{3} - \frac{(-1)^3}{3} = \frac{1}{3} - \left(-\frac{1}{3}\right) = \frac{2}{3}$

Step 5: Calculate the total area

Now, we subtract the results of the two integrals: $\text{Area} = 7.0503 - \frac{2}{3} = 7.0503 - 0.6667 = 6.3836$

Thus, the area of the region bounded by the functions $f(x) = 3e^x - 1$ and $g(x) = x^2 - 1$ on the interval $[-1, 1]$ is approximately $6.384$.

Would you like a detailed explanation on any of the steps? Here are 5 questions to expand on:

1. What happens if the functions intersect more than once on the interval?
2. How do we handle finding intersection points when solving $3e^x = x^2$?
3. Could we estimate the intersection points numerically?
4. How do we integrate exponential and polynomial functions?
5. What if one of the functions was below the x-axis?

Tip: Always check the bounds of integration carefully when working with areas between curves.