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Do not use tRapzedoul. Use the simpsons rule. Round final answer to six decimal places

The problem in the image asks for the evaluation of the integral

\[
\int_0^3 \frac{1}{3 + y^5} \, dy
\]

using **Simpson's Rule** with \(n = 6\). Simpson's Rule is a method of numerical integration, and it is defined as:

\[
\int_a^b f(x) \, dx \approx \frac{h}{3} \left[f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \cdots + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n)\right]
\]

where \( h = \frac{b-a}{n} \) and \( x_i = a + ih \) for \( i = 0, 1, 2, \ldots, n \).

Let's calculate the integral using Simpson's Rule. I will walk through the process:

1. **Determine the step size** \( h \):
   \[
   h = \frac{3 - 0}{6} = 0.5
   \]

2. **Determine the sample points** \( y_i \):
   \[
   y_0 = 0, \, y_1 = 0.5, \, y_2 = 1, \, y_3 = 1.5, \, y_4 = 2, \, y_5 = 2.5, \, y_6 = 3
   \]

3. **Evaluate the function** \( f(y) = \frac{1}{3 + y^5} \) at these points.

4. **Apply Simpson's Rule formula**.

I will now compute the values.Using Simpson's Rule with \(n = 6\), the approximate value of the integral

\[
\int_0^3 \frac{1}{3 + y^5} \, dy
\]

is **0.438575** (rounded to six decimal places).

Would you like any further details or have any questions?

Here are five related questions for further exploration:
1. How does Simpson's Rule compare to the Trapezoidal Rule in terms of accuracy for this problem?
2. What happens to the result if you increase the number of subintervals \(n\)?
3. Can you derive Simpson's Rule from polynomial interpolation?
4. Why does Simpson's Rule require an even number of intervals?
5. How can you estimate the error in Simpson's Rule for a given function?

**Tip:** Simpson's Rule is exact for polynomials of degree three or less, making it especially powerful for smooth, well-behaved functions.

Evaluate the integral ∫(0 to 3) 1 / (3 + y^5) dy using Simpson's Rule with n = 6, rounding the final answer to six decimal places.

Solve the integral ∫(0 to 3) 1 / (3 + y^5) dy using Simpson's Rule with n = 6. This step-by-step solution covers key numerical integration concepts and results in an approximate value of 0.438575, rounded to six decimal places.

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