To find a Trapezoid Rule approximation for the integral of $e^{x^3}$ from 0 to 1 with $n = 100$ subintervals, we can follow these steps:

Trapezoidal Rule Formula

The Trapezoidal Rule for approximating the definite integral $\int_a^b f(x) \, dx$ is given by:

$T_n = \frac{h}{2} \left[ f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_n) \right]$

Where:

• $a$ is the lower limit of integration (in this case, 0),
• $b$ is the upper limit of integration (in this case, 1),
• $h = \frac{b - a}{n}$ is the width of each subinterval,
• $x_i = a + ih$ for $i = 0, 1, 2, \dots, n$.

In this case:

• $a = 0$, $b = 1$,
• $n = 100$,
• $f(x) = e^{x^3}$.

Step-by-Step Calculation:

1. Calculate the step size $h$:

$h = \frac{1 - 0}{100} = 0.01$

2. Calculate the points $x_0, x_1, \dots, x_{100}$:

$x_i = 0 + i \cdot 0.01 = 0.01i \quad \text{for} \quad i = 0, 1, 2, \dots, 100$

3. Compute the function values $f(x_i) = e^{(x_i)^3}$ for $i = 0, 1, 2, \dots, 100$.

4. Apply the Trapezoidal Rule formula:

$T_{100} = \frac{0.01}{2} \left[ f(0) + 2 \sum_{i=1}^{99} f(x_i) + f(1) \right]$

Let me compute this.The Trapezoid Rule approximation for the integral $\int_0^1 e^{x^3} \, dx$ using $n = 100$ subintervals is approximately:

$T_{100} \approx 1.34197$

Would you like more details on the steps, or do you have any further questions?

Here are five related questions to expand your understanding:

1. How does the error in the Trapezoidal Rule decrease as we increase the number of subintervals?
2. What is the actual value of the integral $\int_0^1 e^{x^3} \, dx$?
3. How would the result change if we used Simpson's Rule instead of the Trapezoidal Rule?
4. Can the trapezoidal rule be used for improper integrals?
5. How can you derive the error bound for the Trapezoidal Rule?

Tip: For functions that curve significantly between points, increasing the number of subintervals $n$ can greatly improve the accuracy of the Trapezoidal Rule approximation.