Using the trapezium rule, numerically calculate (to 9
decimal places)
I =
Z 15
−15
e
0.1+0.2x
1
√
2π
e
− x
2
2 dx,
using a step size of
(a) ∆x = 5. I =?

To numerically calculate the integral 

\[
I = \int_{-15}^{15} \frac{e^{0.1 + 0.2x}}{\sqrt{2\pi}} e^{-\frac{x^2}{2}} \, dx
\]

using the **trapezium rule** with a step size of \(\Delta x = 5\), we proceed step by step.

### 1. **Restate the integral**:
We are solving:
\[
I = \int_{-15}^{15} f(x) \, dx
\]
where the function \( f(x) \) is:
\[
f(x) = \frac{e^{0.1 + 0.2x}}{\sqrt{2\pi}} e^{-\frac{x^2}{2}} = \frac{e^{0.1 + 0.2x - \frac{x^2}{2}}}{\sqrt{2\pi}}.
\]

### 2. **Apply the trapezium rule**:
The trapezium rule approximates the integral as:
\[
I \approx \frac{\Delta x}{2} \left[ f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right]
\]
where \( x_0 = -15 \), \( x_n = 15 \), and \( \Delta x = 5 \).

Thus, the points we evaluate the function at are \( x_0 = -15 \), \( x_1 = -10 \), \( x_2 = -5 \), \( x_3 = 0 \), \( x_4 = 5 \), \( x_5 = 10 \), and \( x_6 = 15 \).

### 3. **Calculate the function values**:
Now, we calculate \( f(x) \) at each of these points:

\[
f(x) = \frac{e^{0.1 + 0.2x - \frac{x^2}{2}}}{\sqrt{2\pi}}
\]
- \( f(-15) \)
- \( f(-10) \)
- \( f(-5) \)
- \( f(0) \)
- \( f(5) \)
- \( f(10) \)
- \( f(15) \)

Let's compute these function values and then apply the trapezium rule formula.The approximate value of the integral using the trapezium rule with a step size of \(\Delta x = 5\) is:

\[
I \approx 2.204522385
\]

Would you like further details or clarifications on this method? 

Here are 5 related questions to consider:
1. How would the result change with a smaller step size, like \(\Delta x = 1\)?
2. What is the error bound for the trapezium rule?
3. Can the Simpson’s rule be applied to this integral? How would the results compare?
4. How is the trapezium rule derived, and what are its limitations?
5. Can this integral be solved analytically, and if so, how does the result compare to the numerical solution?

**Tip:** The accuracy of the trapezium rule improves with smaller step sizes but increases computation time. Consider balancing these factors depending on the desired precision.

Using the trapezium rule, numerically calculate (to 9 decimal places) I = ∫ from -15 to 15 of e^(0.1 + 0.2x) / √(2π) * e^(-x²/2) dx, using a step size of (a) ∆x = 5. I =?

Numerically calculate the integral I = ∫ from -15 to 15 of e^(0.1 + 0.2x) / √(2π) * e^(-x²/2) dx using the trapezium rule with a step size of ∆x = 5. This problem demonstrates the use of numerical methods and exponential functions for precise integration.

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