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To estimate the area under the curve from \( x = 0 \) to \( x = 6 \) using both the Trapezoidal Rule and Simpson's Rule, let's go over the details provided and perform the calculations.

### Given Information:
- Function \( y = f(x) \) (as shown in the graph).
- Interval \( [0, 6] \).
- Number of subintervals \( n = 6 \).
- We’ll approximate using:
  - **Trapezoidal Rule** (\( T_6 \))
  - **Simpson's Rule** (\( S_6 \)).

The interval length \( \Delta x \) is:
\[
\Delta x = \frac{6 - 0}{6} = 1
\]

### Part (a): Trapezoidal Rule

The Trapezoidal Rule formula for \( n = 6 \) subintervals is:
\[
T_6 = \frac{\Delta x}{2} \left[ f(x_0) + 2f(x_1) + 2f(x_2) + 2f(x_3) + 2f(x_4) + 2f(x_5) + f(x_6) \right]
\]

### Part (b): Simpson's Rule

The Simpson's Rule formula for \( n = 6 \) is:
\[
S_6 = \frac{\Delta x}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + 2f(x_4) + 4f(x_5) + f(x_6) \right]
\]

### Next Steps

To solve this, I’ll need the approximate values of \( f(x) \) at points \( x = 0, 1, 2, 3, 4, 5, \) and \( 6 \) from the graph. Could you help confirm or estimate these values based on the graph? Or would you like me to explain how to estimate the values visually? 

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Would you like detailed steps once we have the values?

### Related Questions

1. How would the area estimate change if we used more intervals?
2. What’s the difference between Simpson's Rule and the Trapezoidal Rule?
3. How does decreasing the interval width (increasing \( n \)) affect accuracy?
4. Can we apply these methods to functions with sharp peaks or valleys?
5. What other numerical methods could estimate areas under curves?

### Tip
When estimating from graphs, try rounding values to the nearest half or quarter unit to maintain accuracy.

Given the following graph of the function y = f(x) and n = 6, answer the following questions about the area under the curve from x = 0 to x = 6. Use the Trapezoidal Rule to estimate the area (T6) and Simpson's Rule to estimate the area (S6).

Learn how to estimate the area under a curve from x = 0 to x = 6 using the Trapezoidal Rule and Simpson's Rule with n = 6 intervals. Detailed formulas and step-by-step guidance are provided for accurate calculations.

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