To estimate the length of the curve using Simpson's Rule, we'll break it down into steps.

Step 1: Interpret the Curve

We need to estimate the length of the curve defined by the equation:

$L = \int_2^5 \sqrt{16x^2 + 1} \, dx$

Step 2: Apply Simpson's Rule

Simpson's Rule formula for $n$ subintervals is:

$\int_a^b f(x) \, dx \approx \frac{h}{3} \left[ f(x_0) + 4 \sum_{\text{odd } i} f(x_i) + 2 \sum_{\text{even } i} f(x_i) + f(x_n) \right]$

Where:

• $h = \frac{b-a}{n}$ is the width of each subinterval.
• $x_0, x_1, ..., x_n$ are the points between $a$ and $b$.

In our case:

• $a = 2$
• $b = 5$
• $n = 6$

Step 3: Calculate $h$

$h = \frac{5 - 2}{6} = \frac{3}{6} = 0.5$

Step 4: Calculate $x_i$ Values

We will calculate the $x_i$ values for $i = 0, 1, 2, ..., 6$: $x_0 = 2, \quad x_1 = 2.5, \quad x_2 = 3, \quad x_3 = 3.5, \quad x_4 = 4, \quad x_5 = 4.5, \quad x_6 = 5$

Step 5: Compute Function Values

Now, we compute the values of the function $f(x) = \sqrt{16x^2 + 1}$ at these points.

$f(x_0) = \sqrt{16(2)^2 + 1} = \sqrt{64 + 1} = \sqrt{65}$ $f(x_1) = \sqrt{16(2.5)^2 + 1} = \sqrt{100 + 1} = \sqrt{101}$ $f(x_2) = \sqrt{16(3)^2 + 1} = \sqrt{144 + 1} = \sqrt{145}$ $f(x_3) = \sqrt{16(3.5)^2 + 1} = \sqrt{196 + 1} = \sqrt{197}$ $f(x_4) = \sqrt{16(4)^2 + 1} = \sqrt{256 + 1} = \sqrt{257}$ $f(x_5) = \sqrt{16(4.5)^2 + 1} = \sqrt{324 + 1} = \sqrt{325}$ $f(x_6) = \sqrt{16(5)^2 + 1} = \sqrt{400 + 1} = \sqrt{401}$

Step 6: Apply Simpson's Rule Formula

Now, apply Simpson’s Rule: $\int_2^5 \sqrt{16x^2 + 1} \, dx \approx \frac{0.5}{3} \left[ \sqrt{65} + 4(\sqrt{101} + \sqrt{197} + \sqrt{325}) + 2(\sqrt{145} + \sqrt{257}) + \sqrt{401} \right]$

We can compute this directly to get the numerical approximation. Let me calculate this.Using Simpson's Rule with $n = 6$, the estimated length of the curve is approximately $42.114346$ (rounded to six decimal places).

Would you like to go into further details or have any questions about the process?

Here are 5 related questions to consider:

1. How does increasing the number of subintervals $n$ affect the accuracy of Simpson's Rule?
2. What other numerical integration methods could be used to estimate curve lengths?
3. Why does Simpson's Rule require an even number of subintervals?
4. How does the function $\sqrt{16x^2 + 1}$ contribute to the complexity of calculating the curve's length?
5. What is the geometric interpretation of curve length in calculus?

Tip: For better accuracy, especially for complex functions, increasing the number of subintervals in Simpson's Rule can significantly improve the estimate.