To approximate the integral of $\frac{\sin(t)}{t}$ using Simpson's Rule with $n = 4$, we'll first need to recall the general formula for Simpson's Rule:

$\int_a^b f(x) \, dx \approx \frac{h}{3} \left[ f(x_0) + 4 \sum_{i=1,3,5,...,n-1} f(x_i) + 2 \sum_{i=2,4,6,...,n-2} f(x_i) + f(x_n) \right]$

Where:

• $h = \frac{b - a}{n}$ is the step size.
• $n$ is the number of intervals (which is even).
• $x_0, x_1, x_2, \dots, x_n$ are the points on the interval $[a, b]$.

Let's assume you want to integrate $\frac{\sin(t)}{t}$ over the interval $[a, b]$. For simplicity, let's take the standard interval $[0, 1]$. We need to use Simpson's Rule for this integral.

1. Define the function: $f(t) = \frac{\sin(t)}{t}$. Note that we must handle $t = 0$ carefully since $\frac{\sin(0)}{0}$ is undefined. However, we know from calculus that $\lim_{t \to 0} \frac{\sin(t)}{t} = 1$, so we can evaluate it as 1 at $t = 0$.

2. Set $n = 4$, so $h = \frac{1 - 0}{4} = 0.25$.

3. Evaluate the points: We need to compute $f(t)$ at $t = 0, 0.25, 0.5, 0.75, 1$:

   • $f(0) = 1$
   • $f(0.25) = \frac{\sin(0.25)}{0.25}$
   • $f(0.5) = \frac{\sin(0.5)}{0.5}$
   • $f(0.75) = \frac{\sin(0.75)}{0.75}$
   • $f(1) = \frac{\sin(1)}{1}$

4. Apply Simpson's Rule:

$I \approx \frac{0.25}{3} \left[ f(0) + 4(f(0.25) + f(0.5) + f(0.75)) + f(1) \right]$

Now, we can compute the numerical values for each of these functions:

• $f(0.25) \approx \frac{\sin(0.25)}{0.25} \approx 1.0032$
• $f(0.5) \approx \frac{\sin(0.5)}{0.5} \approx 0.9589$
• $f(0.75) \approx \frac{\sin(0.75)}{0.75} \approx 0.9732$
• $f(1) \approx \frac{\sin(1)}{1} \approx 0.8415$

So,

$I \approx \frac{0.25}{3} \left[ 1 + 4(1.0032 + 0.9589 + 0.9732) + 0.8415 \right]$

Now, simplifying:

$I \approx \frac{0.25}{3} \left[ 1 + 4(2.9353) + 0.8415 \right]$

$I \approx \frac{0.25}{3} \left[ 1 + 11.7412 + 0.8415 \right]$

$I \approx \frac{0.25}{3} \left[ 13.5827 \right]$

$I \approx \frac{3.3957}{3} \approx 1.1319$

So, the approximate integral value is 1.1319.

Would you like more details or help with any other question? Here are a few related questions you might find interesting:

1. How does Simpson's Rule compare to other numerical integration methods like the trapezoidal rule?
2. What happens when you use a higher value for $n$ in Simpson's Rule?
3. How can Simpson’s Rule be modified for non-uniform intervals?
4. How do we deal with the singularity at $t = 0$ for functions like $\frac{\sin(t)}{t}$?
5. What is the effect of increasing the number of intervals in improving accuracy?

Tip: When using Simpson’s Rule, the accuracy improves significantly with an increase in $n$, but keep in mind that $n$ must always be an even number.