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Sagot :
To find the approximations [tex]\( T_{10} \)[/tex], [tex]\( M_{10} \)[/tex], and [tex]\( S_{10} \)[/tex] for the integral [tex]\(\int_0^\pi 11 \sin (x) \, dx\)[/tex], we will use the Trapezoidal Rule, the Midpoint Rule, and Simpson's Rule with [tex]\( n = 10 \)[/tex] subintervals. Let’s break down each rule and calculate the approximations.
### 1. The Trapezoidal Rule ([tex]\( T_{10} \)[/tex])
The Trapezoidal Rule approximates the integral by summing the areas of trapezoids under the curve. The formula is given by:
[tex]\[ T_n = \frac{b - a}{2n} \left[f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n)\right] \][/tex]
For [tex]\( n = 10 \)[/tex], [tex]\( a = 0 \)[/tex], and [tex]\( b = \pi \)[/tex]:
- The width of each subinterval ([tex]\( \Delta x \)[/tex]) is [tex]\(\frac{\pi}{10}\)[/tex].
- Calculate the function values at each subinterval point.
### 2. The Midpoint Rule ([tex]\( M_{10} \)[/tex])
The Midpoint Rule uses the midpoint of each interval to approximate the area. The formula is:
[tex]\[ M_n = \frac{b - a}{n} \sum_{i=1}^{n} f\left(x_{i-\frac{1}{2}}\right) \][/tex]
For [tex]\( n = 10 \)[/tex], [tex]\( a = 0 \)[/tex], and [tex]\( b = \pi \)[/tex]:
- The width of each subinterval ([tex]\( \Delta x \)[/tex]) is [tex]\(\frac{\pi}{10}\)[/tex].
- Calculate the function values at the midpoints of each subinterval.
### 3. Simpson's Rule ([tex]\( S_{10} \)[/tex])
Simpson's Rule is a more accurate method that approximates the integral by a combination of parabolic segments:
[tex]\[ S_n = \frac{\Delta x}{3} \left[f(x_0) + 4 \sum_{i \text{ odd}} f(x_i) + 2 \sum_{i \text{ even}} f(x_i) + f(x_n)\right] \][/tex]
For [tex]\( n = 10 \)[/tex], [tex]\( a = 0 \)[/tex], and [tex]\( b = \pi \)[/tex]:
- The width of each subinterval ([tex]\( \Delta x \)[/tex]) is [tex]\(\frac{\pi}{10}\)[/tex].
- Calculate the function values at each subinterval point, giving more importance to the midpoints between each even and odd subinterval.
After performing the calculations for [tex]\( T_{10} \)[/tex], [tex]\( M_{10} \)[/tex], and [tex]\( S_{10} \)[/tex], the approximations to five decimal places are:
[tex]\[ \begin{array}{l} T_{10} = 21.818759 \\ M_{10} = 22.090732 \\ S_{10} = 22.001205 \end{array} \][/tex]
Thus, the approximations are:
[tex]\[ \begin{array}{l} T_{10} = 21.818759 \\ M_{10} = 22.090732 \\ S_{10} = 22.001205 \end{array} \][/tex]
### 1. The Trapezoidal Rule ([tex]\( T_{10} \)[/tex])
The Trapezoidal Rule approximates the integral by summing the areas of trapezoids under the curve. The formula is given by:
[tex]\[ T_n = \frac{b - a}{2n} \left[f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n)\right] \][/tex]
For [tex]\( n = 10 \)[/tex], [tex]\( a = 0 \)[/tex], and [tex]\( b = \pi \)[/tex]:
- The width of each subinterval ([tex]\( \Delta x \)[/tex]) is [tex]\(\frac{\pi}{10}\)[/tex].
- Calculate the function values at each subinterval point.
### 2. The Midpoint Rule ([tex]\( M_{10} \)[/tex])
The Midpoint Rule uses the midpoint of each interval to approximate the area. The formula is:
[tex]\[ M_n = \frac{b - a}{n} \sum_{i=1}^{n} f\left(x_{i-\frac{1}{2}}\right) \][/tex]
For [tex]\( n = 10 \)[/tex], [tex]\( a = 0 \)[/tex], and [tex]\( b = \pi \)[/tex]:
- The width of each subinterval ([tex]\( \Delta x \)[/tex]) is [tex]\(\frac{\pi}{10}\)[/tex].
- Calculate the function values at the midpoints of each subinterval.
### 3. Simpson's Rule ([tex]\( S_{10} \)[/tex])
Simpson's Rule is a more accurate method that approximates the integral by a combination of parabolic segments:
[tex]\[ S_n = \frac{\Delta x}{3} \left[f(x_0) + 4 \sum_{i \text{ odd}} f(x_i) + 2 \sum_{i \text{ even}} f(x_i) + f(x_n)\right] \][/tex]
For [tex]\( n = 10 \)[/tex], [tex]\( a = 0 \)[/tex], and [tex]\( b = \pi \)[/tex]:
- The width of each subinterval ([tex]\( \Delta x \)[/tex]) is [tex]\(\frac{\pi}{10}\)[/tex].
- Calculate the function values at each subinterval point, giving more importance to the midpoints between each even and odd subinterval.
After performing the calculations for [tex]\( T_{10} \)[/tex], [tex]\( M_{10} \)[/tex], and [tex]\( S_{10} \)[/tex], the approximations to five decimal places are:
[tex]\[ \begin{array}{l} T_{10} = 21.818759 \\ M_{10} = 22.090732 \\ S_{10} = 22.001205 \end{array} \][/tex]
Thus, the approximations are:
[tex]\[ \begin{array}{l} T_{10} = 21.818759 \\ M_{10} = 22.090732 \\ S_{10} = 22.001205 \end{array} \][/tex]
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