To find the area enclosed by the curve $y = e^{x^2}$ and the x-axis from $x = -1$ to $x = 1$, using the trapezoidal rule, we first need to apply the trapezoidal approximation formula:

$A \approx \frac{h}{2} \left[ f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right]$

Where:

• $f(x)$ is the function, in this case $f(x) = e^{x^2}$,
• $x_0 = -1$ and $x_n = 1$ are the bounds of integration,
• $h$ is the width of each subinterval, and
• $n$ is the number of subintervals.

Step 1: Choose the number of subintervals

Let's take $n = 4$ subintervals (you can adjust this for better accuracy). The width of each subinterval $h$ is given by:

$h = \frac{x_n - x_0}{n} = \frac{1 - (-1)}{4} = \frac{2}{4} = 0.5$

Step 2: Calculate the function values

We need to evaluate $f(x) = e^{x^2}$ at the points $x_0, x_1, x_2, x_3, x_4$:

• $x_0 = -1$, so $f(-1) = e^{(-1)^2} = e^1 \approx 2.718$,
• $x_1 = -0.5$, so $f(-0.5) = e^{(-0.5)^2} = e^{0.25} \approx 1.284$,
• $x_2 = 0$, so $f(0) = e^{0^2} = e^0 = 1$,
• $x_3 = 0.5$, so $f(0.5) = e^{(0.5)^2} = e^{0.25} \approx 1.284$,
• $x_4 = 1$, so $f(1) = e^{(1)^2} = e^1 \approx 2.718$.

Step 3: Apply the trapezoidal rule

Using the trapezoidal rule formula:

$A \approx \frac{h}{2} \left[ f(-1) + 2(f(-0.5) + f(0.5)) + f(1) \right]$

Substitute the values:

$A \approx \frac{0.5}{2} \left[ 2.718 + 2(1.284 + 1.284) + 2.718 \right]$

$A \approx 0.25 \left[ 2.718 + 2(2.568) + 2.718 \right]$

$A \approx 0.25 \left[ 2.718 + 5.136 + 2.718 \right]$

$A \approx 0.25 \times 10.572$

$A \approx 2.643$

Final Answer:

The area enclosed by the curve and the x-axis from $x = -1$ to $x = 1$ using the trapezoidal rule is approximately 2.64 (rounded to two decimal places).

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Would you like to explore the method further or have any questions?

Related questions to deepen understanding:

1. How does the trapezoidal rule compare to Simpson's rule for approximating the area?
2. What happens to the accuracy of the trapezoidal rule as the number of subintervals increases?
3. How would you apply the trapezoidal rule with a larger number of subintervals (say, 10)?
4. What is the exact integral of $e^{x^2}$ from $x = -1$ to $x = 1$, if it can be evaluated analytically?
5. What are the benefits of using numerical integration methods like the trapezoidal rule?

Tip: The trapezoidal rule improves in accuracy as you increase the number of subintervals. If you want to refine your result, you can increase $n$ or use Simpson's rule for better precision.