The instructions for the given integral have two​ parts, one for the trapezoidal rule and one for​ Simpson's rule. Complete the following parts.
Integral from 0 to pi 3 sine t font size decreased by 5 dt
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Part 1
I. Using the trapezoidal rule to estimate the integral with nequals4 steps
Tequals
  

​(Simplify your answer. Round to four decimal places as​ needed.)
II. Evaluate the integral directly.
Integral from 0 to pi 3 sine font size decreased by 5 dtequals
  

​(Type an integer or a​ decimal.)
III. Using​ Simpson's rule to estimate the integral with nequals4 steps
Upper Sequals
  

​(Simplify your answer. Round to four decimal places as​ needed.)
Brainly Ginny character avatar


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.

The instructions for the given integral have two​ parts, one for the trapezoidal rule and one for​ Simpson's rule. Complete the following parts. Integral from 0 to pi 3 sine t dt: I. Using the trapezoidal rule to estimate the integral with n = 4 steps. II. Evaluate the integral directly. III. Using​ Simpson's rule to estimate the integral with n = 4 steps.

Learn how to estimate the integral of 3sin(t) from 0 to π using the trapezoidal rule and Simpson's rule with 4 steps. Compare these approximations with the exact value of the integral, applying key numerical integration techniques. Suitable for students in Grades 11-12 and early college.

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