We are tasked with approximating the integral of $f(x)$ from $0$ to $\pi$ using numerical methods: the Trapezoidal Rule ($T_n$), Midpoint Rule ($M_n$), and Simpson's Rule ($S_n$). Additionally, we want to ensure that the error for each approximation is within $0.00001$ or $10^{-5}$, given bounds on the second and fourth derivatives of the function $f(x)$.

Given:

• $|f''(x)| \leq 39$
• $|f^{(4)}(x)| \leq 39$
• The interval of integration is $[0, \pi]$.

1. Trapezoidal Rule Error:

The error for the Trapezoidal Rule is given by:

$E_T = \frac{(\pi - 0)^3}{12n^2} \cdot \max_{x \in [0, \pi]} |f''(x)|$

Substituting the given values:

$E_T \leq \frac{\pi^3}{12n^2} \cdot 39$

We want $E_T \leq 10^{-5}$, so we solve:

$\frac{\pi^3 \cdot 39}{12n^2} \leq 10^{-5}$

$n^2 \geq \frac{\pi^3 \cdot 39}{12 \cdot 10^{-5}}$

$n \geq \sqrt{\frac{\pi^3 \cdot 39}{12 \cdot 10^{-5}}}$

2. Midpoint Rule Error:

The error for the Midpoint Rule is similar to the Trapezoidal Rule:

$E_M = \frac{(\pi - 0)^3}{24n^2} \cdot \max_{x \in [0, \pi]} |f''(x)|$

Substituting values:

$E_M \leq \frac{\pi^3}{24n^2} \cdot 39$

We want $E_M \leq 10^{-5}$, so we solve:

$\frac{\pi^3 \cdot 39}{24n^2} \leq 10^{-5}$

$n^2 \geq \frac{\pi^3 \cdot 39}{24 \cdot 10^{-5}}$

$n \geq \sqrt{\frac{\pi^3 \cdot 39}{24 \cdot 10^{-5}}}$

3. Simpson's Rule Error:

The error for Simpson's Rule is given by:

$E_S = \frac{(\pi - 0)^5}{180n^4} \cdot \max_{x \in [0, \pi]} |f^{(4)}(x)|$

Substituting the values:

$E_S \leq \frac{\pi^5}{180n^4} \cdot 39$

We want $E_S \leq 10^{-5}$, so we solve:

$\frac{\pi^5 \cdot 39}{180n^4} \leq 10^{-5}$

$n^4 \geq \frac{\pi^5 \cdot 39}{180 \cdot 10^{-5}}$

$n \geq \sqrt[4]{\frac{\pi^5 \cdot 39}{180 \cdot 10^{-5}}}$

Summary of $n$ values:

1. $n \geq \sqrt{\frac{\pi^3 \cdot 39}{12 \cdot 10^{-5}}}$ for the Trapezoidal Rule
2. $n \geq \sqrt{\frac{\pi^3 \cdot 39}{24 \cdot 10^{-5}}}$ for the Midpoint Rule
3. $n \geq \sqrt[4]{\frac{\pi^5 \cdot 39}{180 \cdot 10^{-5}}}$ for Simpson's Rule

Now, let's compute these values.To ensure that the approximations $T_n$, $M_n$, and $S_n$ are accurate to within $0.00001$, the values of $n$ must be chosen as follows:

1. For the Trapezoidal Rule: $n \geq 3175$
2. For the Midpoint Rule: $n \geq 2245$
3. For Simpson's Rule: $n \geq 51$

Simpson's Rule requires a significantly smaller $n$ for the same level of accuracy compared to the Trapezoidal and Midpoint Rules.

Would you like further details on these methods or any clarifications?

Related Questions:

1. How do we derive error bounds for these numerical methods?
2. What factors influence the accuracy of these numerical approximations?
3. Can these methods be used for improper integrals?
4. How do you improve accuracy without increasing $n$?
5. How does the function's behavior affect the choice of numerical method?

Tip:

Simpson's Rule often provides better accuracy with fewer intervals than both the Trapezoidal and Midpoint Rules, especially when the integrand is smooth.