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Enter the function [tex]\( f(x) \)[/tex], interval [tex]\([a, b]\)[/tex], and number of subintervals (rectangles) [tex]\( n \)[/tex] below. [tex]\( R_n \)[/tex] is the area sum for right endpoints and [tex]\( L_n \)[/tex] is the area sum for left endpoints. The example given below is for [tex]\( f(x)=5 \cos (x) \)[/tex], [tex]\([0, \pi / 2] \)[/tex], [tex]\( a=0 \)[/tex], [tex]\( b=\pi / 2 \)[/tex], and [tex]\( n=10 \)[/tex].

Example:
[tex]\[
\begin{aligned}
f(x) &= 5 \cos (x) \\
a &= 0 \\
b &= \frac{\pi}{2} \quad (b \approx 1.570796326) \\
n &= 10
\end{aligned}
\][/tex]

Formulas:
[tex]\[
R_n = \sum_{i=1}^n f\left(a + i \cdot \frac{b-a}{n}\right) \cdot \frac{b-a}{n}
\][/tex]

[tex]\[
L_n = \sum_{i=0}^{n-1} f\left(a + i \cdot \frac{b-a}{n}\right) \cdot \frac{b-a}{n}
\][/tex]

You can edit any or all of these.

Sagot :

GinnyAnsweridn
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.
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