Let's work through each part of this problem step by step. The integral we are approximating and calculating is:

$I = \int_0^\pi 3 \sin(t) \, dt$

Part I: Trapezoidal Rule Approximation

The trapezoidal rule formula for approximating an integral $\int_a^b f(x) dx$ with $n$ steps is given by:

$T_n = \frac{b - a}{2n} \left( f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right)$

Here:

• $a = 0$, $b = \pi$
• $f(t) = 3 \sin(t)$
• $n = 4$, so we divide the interval $[0, \pi]$ into 4 equal subintervals, each of width $h = \frac{\pi - 0}{4} = \frac{\pi}{4}$.

We now calculate the values at each point $x_i = a + ih$, where $i = 0, 1, 2, 3, 4$:

• $x_0 = 0$
• $x_1 = \frac{\pi}{4}$
• $x_2 = \frac{\pi}{2}$
• $x_3 = \frac{3\pi}{4}$
• $x_4 = \pi$

The function values are:

• $f(x_0) = 3\sin(0) = 0$
• $f(x_1) = 3\sin\left(\frac{\pi}{4}\right) = 3 \times \frac{\sqrt{2}}{2} \approx 2.1213$
• $f(x_2) = 3\sin\left(\frac{\pi}{2}\right) = 3$
• $f(x_3) = 3\sin\left(\frac{3\pi}{4}\right) = 3 \times \frac{\sqrt{2}}{2} \approx 2.1213$
• $f(x_4) = 3\sin(\pi) = 0$

Now, apply the trapezoidal rule formula: $T_4 = \frac{\pi}{8} \left( 0 + 2(2.1213 + 3 + 2.1213) + 0 \right)$ $T_4 = \frac{\pi}{8} \times 2(7.2426) = \frac{\pi}{8} \times 14.4852$ $T_4 \approx 5.698$

Part II: Direct Evaluation of the Integral

We can directly evaluate the integral $\int_0^\pi 3 \sin(t) dt$. The integral of $3 \sin(t)$ is:

$\int 3 \sin(t) dt = -3 \cos(t)$

Now, evaluate it from $0$ to $\pi$:

$\left[ -3 \cos(t) \right]_0^\pi = -3 \cos(\pi) + 3 \cos(0)$ $= -3(-1) + 3(1) = 3 + 3 = 6$

Thus, the exact value of the integral is:

$\int_0^\pi 3 \sin(t) dt = 6$

Part III: Simpson's Rule Approximation

The Simpson's rule formula for approximating an integral $\int_a^b f(x) dx$ with $n$ steps (where $n$ must be even) is given by:

$S_n = \frac{b - a}{3n} \left( f(x_0) + 4 \sum_{\text{odd } i} f(x_i) + 2 \sum_{\text{even } i} f(x_i) + f(x_n) \right)$

Here:

• $n = 4$, and the points $x_0, x_1, x_2, x_3, x_4$ are the same as in the trapezoidal rule part.
• Function values are also the same.

Applying Simpson’s rule: $S_4 = \frac{\pi}{12} \left( 0 + 4(2.1213 + 2.1213) + 2(3) + 0 \right)$ $S_4 = \frac{\pi}{12} \left( 0 + 4 \times 4.2426 + 6 + 0 \right)$ $S_4 = \frac{\pi}{12} \times (16.9704 + 6) = \frac{\pi}{12} \times 22.9704$ $S_4 \approx 6.0214$

Final Results:

1. Trapezoidal Rule Estimate: $T_4 \approx 5.698$
2. Exact Value of the Integral: 6
3. Simpson's Rule Estimate: $S_4 \approx 6.0214$

Do you want more details or explanations on any of the steps?

Here are 5 related questions to extend your understanding:

1. How does the trapezoidal rule converge compared to Simpson's rule?
2. What happens to the approximation if you increase the number of subintervals?
3. Can Simpson's rule be applied with an odd number of steps?
4. How would you adapt these methods for a different function, like $\cos(t)$?
5. What is the geometric interpretation of the trapezoidal and Simpson's rule?

Tip: Increasing the number of steps generally improves the accuracy of numerical integration methods like the trapezoidal and Simpson’s rule.