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Sagot :
Certainly! Let's work through this problem step-by-step. The task is to find the Riemann sums using right endpoints and left endpoints for the given function [tex]\( f(x) = 5 \cos x \)[/tex] over the interval [tex]\([a, b] = [0, \frac{\pi}{2}]\)[/tex] with [tex]\( n = 10 \)[/tex] subintervals.
### Step-by-Step Solution
1. Define the Function and Interval:
- [tex]\( f(x) = 5 \cos x \)[/tex]
- [tex]\( a = 0 \)[/tex]
- [tex]\( b = \frac{\pi}{2} \approx 1.570796326 \)[/tex]
2. Determine the Number of Subintervals:
- [tex]\( n = 10 \)[/tex]
3. Calculate the Width of Each Subinterval:
[tex]\[ \Delta x = \frac{b - a}{n} = \frac{\frac{\pi}{2} - 0}{10} = \frac{\pi}{20} \approx 0.15707963267948966 \][/tex]
4. Calculate the Area Sum for Right Endpoints ( [tex]\( R_n \)[/tex] ):
- The formula for the Riemann sum using right endpoints is:
[tex]\[ R_n = \sum_{i=1}^{n} f \left( a + i \cdot \Delta x \right) \cdot \Delta x \][/tex]
- Here, [tex]\( a + i \cdot \Delta x \)[/tex] gives the right-endpoint [tex]\( x \)[/tex]-coordinate for each subinterval.
- Substituting the values and summing up:
[tex]\[ R_n = f(0.15707963267948966) \Delta x + f(0.3141592653589793) \Delta x + \cdots + f(1.5707963267948966) \Delta x \][/tex]
- With the actual calculations, the result is:
[tex]\[ R_n \approx 4.597015850073062 \][/tex]
5. Calculate the Area Sum for Left Endpoints ( [tex]\( L_n \)[/tex] ):
- The formula for the Riemann sum using left endpoints is:
[tex]\[ L_n = \sum_{i=0}^{n-1} f \left( a + i \cdot \Delta x \right) \cdot \Delta x \][/tex]
- Here, [tex]\( a + i \cdot \Delta x \)[/tex] gives the left-endpoint [tex]\( x \)[/tex]-coordinate for each subinterval.
- Substituting the values and summing up:
[tex]\[ L_n = f(0) \Delta x + f(0.15707963267948966) \Delta x + \cdots + f(1.413716694115407) \Delta x \][/tex]
- With the actual calculations, the result is:
[tex]\[ L_n \approx 5.38241401347051 \][/tex]
### Summary of Results
- The width of each subinterval ([tex]\( \Delta x \)[/tex]) is approximately [tex]\( 0.15707963267948966 \)[/tex].
- The area sum for right endpoints ([tex]\( R_n \)[/tex]) is approximately [tex]\( 4.597015850073062 \)[/tex].
- The area sum for left endpoints ([tex]\( L_n \)[/tex]) is approximately [tex]\( 5.38241401347051 \)[/tex].
These results give us a good approximation of the integral of [tex]\( f(x) = 5 \cos x \)[/tex] over the interval [tex]\([0, \frac{\pi}{2}]\)[/tex] using both right and left Riemann sums with [tex]\( n = 10 \)[/tex] subintervals.
### Step-by-Step Solution
1. Define the Function and Interval:
- [tex]\( f(x) = 5 \cos x \)[/tex]
- [tex]\( a = 0 \)[/tex]
- [tex]\( b = \frac{\pi}{2} \approx 1.570796326 \)[/tex]
2. Determine the Number of Subintervals:
- [tex]\( n = 10 \)[/tex]
3. Calculate the Width of Each Subinterval:
[tex]\[ \Delta x = \frac{b - a}{n} = \frac{\frac{\pi}{2} - 0}{10} = \frac{\pi}{20} \approx 0.15707963267948966 \][/tex]
4. Calculate the Area Sum for Right Endpoints ( [tex]\( R_n \)[/tex] ):
- The formula for the Riemann sum using right endpoints is:
[tex]\[ R_n = \sum_{i=1}^{n} f \left( a + i \cdot \Delta x \right) \cdot \Delta x \][/tex]
- Here, [tex]\( a + i \cdot \Delta x \)[/tex] gives the right-endpoint [tex]\( x \)[/tex]-coordinate for each subinterval.
- Substituting the values and summing up:
[tex]\[ R_n = f(0.15707963267948966) \Delta x + f(0.3141592653589793) \Delta x + \cdots + f(1.5707963267948966) \Delta x \][/tex]
- With the actual calculations, the result is:
[tex]\[ R_n \approx 4.597015850073062 \][/tex]
5. Calculate the Area Sum for Left Endpoints ( [tex]\( L_n \)[/tex] ):
- The formula for the Riemann sum using left endpoints is:
[tex]\[ L_n = \sum_{i=0}^{n-1} f \left( a + i \cdot \Delta x \right) \cdot \Delta x \][/tex]
- Here, [tex]\( a + i \cdot \Delta x \)[/tex] gives the left-endpoint [tex]\( x \)[/tex]-coordinate for each subinterval.
- Substituting the values and summing up:
[tex]\[ L_n = f(0) \Delta x + f(0.15707963267948966) \Delta x + \cdots + f(1.413716694115407) \Delta x \][/tex]
- With the actual calculations, the result is:
[tex]\[ L_n \approx 5.38241401347051 \][/tex]
### Summary of Results
- The width of each subinterval ([tex]\( \Delta x \)[/tex]) is approximately [tex]\( 0.15707963267948966 \)[/tex].
- The area sum for right endpoints ([tex]\( R_n \)[/tex]) is approximately [tex]\( 4.597015850073062 \)[/tex].
- The area sum for left endpoints ([tex]\( L_n \)[/tex]) is approximately [tex]\( 5.38241401347051 \)[/tex].
These results give us a good approximation of the integral of [tex]\( f(x) = 5 \cos x \)[/tex] over the interval [tex]\([0, \frac{\pi}{2}]\)[/tex] using both right and left Riemann sums with [tex]\( n = 10 \)[/tex] subintervals.
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