We are asked to approximate the integral of $\int_5^7 \frac{\sin(t)}{t} \, dt$ using three numerical methods: the Trapezoidal Rule, the Midpoint Rule, and Simpson's Rule, with $n = 4$. Let's go through each method step by step.

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Step 1: Set up the problem

• Given function: $f(t) = \frac{\sin(t)}{t}$
• Interval: $[5, 7]$
• Number of subintervals: $n = 4$

The width of each subinterval is calculated as:

$h = \frac{b - a}{n} = \frac{7 - 5}{4} = 0.5$

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Step 2: (a) Trapezoidal Rule

The Trapezoidal Rule is given by the formula:

$T_n = \frac{h}{2} \left[ f(a) + 2 \sum_{i=1}^{n-1} f(x_i) + f(b) \right]$

For this integral:

• $a = 5$, $b = 7$, $h = 0.5$
• $x_0 = 5$, $x_1 = 5.5$, $x_2 = 6$, $x_3 = 6.5$, $x_4 = 7$

Let's now calculate the values of $f(t) = \frac{\sin(t)}{t}$ at these points.

Step 3: (b) Midpoint Rule

The Midpoint Rule is given by the formula:

$M_n = h \sum_{i=1}^{n} f\left( \frac{x_{i-1} + x_i}{2} \right)$

Here, the midpoints are:

• Midpoints: $5.25$, $5.75$, $6.25$, $6.75$

We will evaluate $f(t) = \frac{\sin(t)}{t}$ at these points.

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Step 4: (c) Simpson's Rule

Simpson's Rule is given by:

$S_n = \frac{h}{3} \left[ f(a) + 4 \sum_{i=1,3,\dots}^{n-1} f(x_i) + 2 \sum_{i=2,4,\dots}^{n-2} f(x_i) + f(b) \right]$

We will also need to calculate the values of $f(t)$ at the same points as in the Trapezoidal Rule.

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Now, let's proceed with the calculations for each method.Here are the approximations for the integral $\int_5^7 \frac{\sin(t)}{t} \, dt$ using the specified numerical methods with $n = 4$:

• (a) Trapezoidal Rule: $-0.095359$
• (b) Midpoint Rule: $-0.095319$
• (c) Simpson's Rule: $-0.095301$

Would you like more details or have any questions?

Here are five related questions for further exploration:

1. How does the accuracy of the Trapezoidal Rule compare to Simpson's Rule for different values of $n$?
2. What is the geometric interpretation of each numerical method?
3. How would the results change if the function had a discontinuity in the interval?
4. Can these methods be applied to improper integrals, and if so, how?
5. What are the error bounds for each rule, and how do they depend on $n$?

Tip: Simpson's Rule is generally more accurate than both the Trapezoidal and Midpoint Rules, especially for smooth functions.