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Sagot :
Let's find the left and right Riemann sums for the function [tex]\( f(x) = \frac{3}{x} + 2 \)[/tex] on the interval [tex]\([1, 5]\)[/tex] with [tex]\( n = 4 \)[/tex].
### Step-by-Step Solution:
1. Determine [tex]\(\Delta x\)[/tex]:
[tex]\[ \Delta x = \frac{b - a}{n} = \frac{5 - 1}{4} = 1 \][/tex]
2. Left Riemann Sum:
The left Riemann sum uses the left endpoints of each subinterval. The points we will evaluate [tex]\( f(x) \)[/tex] at are [tex]\( x_0, x_1, x_2, \)[/tex] and [tex]\( x_3 \)[/tex], where:
[tex]\[ x_0 = a = 1, \quad x_1 = 2, \quad x_2 = 3, \quad x_3 = 4 \][/tex]
The left Riemann sum is given by:
[tex]\[ L_n = \sum_{i=0}^{n-1} f(x_i) \Delta x \][/tex]
Substitute [tex]\( f(x) \)[/tex] and the values of [tex]\( x \)[/tex]:
[tex]\[ L_4 = \left(f(1) + f(2) + f(3) + f(4)\right) \Delta x \][/tex]
Calculate each [tex]\( f(x) \)[/tex]:
[tex]\[ f(1) = \frac{3}{1} + 2 = 5 \][/tex]
[tex]\[ f(2) = \frac{3}{2} + 2 = \frac{7}{2} \][/tex]
[tex]\[ f(3) = \frac{3}{3} + 2 = 3 \][/tex]
[tex]\[ f(4) = \frac{3}{4} + 2 = \frac{11}{4} \][/tex]
Now, sum these values and multiply by [tex]\(\Delta x\)[/tex]:
[tex]\[ L_4 = \left( 5 + \frac{7}{2} + 3 + \frac{11}{4} \right) \cdot 1 \][/tex]
Simplify the sum:
[tex]\[ L_4 = 5 + \frac{7}{2} + 3 + \frac{11}{4} = \frac{20}{4} + \frac{14}{4} + \frac{12}{4} + \frac{11}{4} = \frac{57}{4} \][/tex]
Therefore, the left Riemann sum is:
[tex]\[ \boxed{\frac{57}{4}} \][/tex]
3. Right Riemann Sum:
The right Riemann sum uses the right endpoints of each subinterval. The points we will evaluate [tex]\( f(x) \)[/tex] at are [tex]\( x_1, x_2, x_3, \)[/tex] and [tex]\( x_4 \)[/tex], where:
[tex]\[ x_1 = 2, \quad x_2 = 3, \quad x_3 = 4, \quad x_4 = b = 5 \][/tex]
The right Riemann sum is given by:
[tex]\[ R_n = \sum_{i=1}^{n} f(x_i) \Delta x \][/tex]
Substitute [tex]\( f(x) \)[/tex] and the values of [tex]\( x \)[/tex]:
[tex]\[ R_4 = \left(f(2) + f(3) + f(4) + f(5)\right) \Delta x \][/tex]
Calculate [tex]\( f(5) \)[/tex]:
[tex]\[ f(5) = \frac{3}{5} + 2 = \frac{13}{5} \][/tex]
Now, sum these values and multiply by [tex]\(\Delta x\)[/tex]:
[tex]\[ R_4 = \left( \frac{7}{2} + 3 + \frac{11}{4} + \frac{13}{5} \right) \cdot 1 \][/tex]
Simplify the sum (by normalizing to a common denominator, if needed) and add the values:
(We'll use the Python-computed value directly to avoid manual errors)
[tex]\[ R_4 = 11.85 \][/tex]
Therefore, the right Riemann sum is:
[tex]\[ \boxed{11.85} \][/tex]
### Step-by-Step Solution:
1. Determine [tex]\(\Delta x\)[/tex]:
[tex]\[ \Delta x = \frac{b - a}{n} = \frac{5 - 1}{4} = 1 \][/tex]
2. Left Riemann Sum:
The left Riemann sum uses the left endpoints of each subinterval. The points we will evaluate [tex]\( f(x) \)[/tex] at are [tex]\( x_0, x_1, x_2, \)[/tex] and [tex]\( x_3 \)[/tex], where:
[tex]\[ x_0 = a = 1, \quad x_1 = 2, \quad x_2 = 3, \quad x_3 = 4 \][/tex]
The left Riemann sum is given by:
[tex]\[ L_n = \sum_{i=0}^{n-1} f(x_i) \Delta x \][/tex]
Substitute [tex]\( f(x) \)[/tex] and the values of [tex]\( x \)[/tex]:
[tex]\[ L_4 = \left(f(1) + f(2) + f(3) + f(4)\right) \Delta x \][/tex]
Calculate each [tex]\( f(x) \)[/tex]:
[tex]\[ f(1) = \frac{3}{1} + 2 = 5 \][/tex]
[tex]\[ f(2) = \frac{3}{2} + 2 = \frac{7}{2} \][/tex]
[tex]\[ f(3) = \frac{3}{3} + 2 = 3 \][/tex]
[tex]\[ f(4) = \frac{3}{4} + 2 = \frac{11}{4} \][/tex]
Now, sum these values and multiply by [tex]\(\Delta x\)[/tex]:
[tex]\[ L_4 = \left( 5 + \frac{7}{2} + 3 + \frac{11}{4} \right) \cdot 1 \][/tex]
Simplify the sum:
[tex]\[ L_4 = 5 + \frac{7}{2} + 3 + \frac{11}{4} = \frac{20}{4} + \frac{14}{4} + \frac{12}{4} + \frac{11}{4} = \frac{57}{4} \][/tex]
Therefore, the left Riemann sum is:
[tex]\[ \boxed{\frac{57}{4}} \][/tex]
3. Right Riemann Sum:
The right Riemann sum uses the right endpoints of each subinterval. The points we will evaluate [tex]\( f(x) \)[/tex] at are [tex]\( x_1, x_2, x_3, \)[/tex] and [tex]\( x_4 \)[/tex], where:
[tex]\[ x_1 = 2, \quad x_2 = 3, \quad x_3 = 4, \quad x_4 = b = 5 \][/tex]
The right Riemann sum is given by:
[tex]\[ R_n = \sum_{i=1}^{n} f(x_i) \Delta x \][/tex]
Substitute [tex]\( f(x) \)[/tex] and the values of [tex]\( x \)[/tex]:
[tex]\[ R_4 = \left(f(2) + f(3) + f(4) + f(5)\right) \Delta x \][/tex]
Calculate [tex]\( f(5) \)[/tex]:
[tex]\[ f(5) = \frac{3}{5} + 2 = \frac{13}{5} \][/tex]
Now, sum these values and multiply by [tex]\(\Delta x\)[/tex]:
[tex]\[ R_4 = \left( \frac{7}{2} + 3 + \frac{11}{4} + \frac{13}{5} \right) \cdot 1 \][/tex]
Simplify the sum (by normalizing to a common denominator, if needed) and add the values:
(We'll use the Python-computed value directly to avoid manual errors)
[tex]\[ R_4 = 11.85 \][/tex]
Therefore, the right Riemann sum is:
[tex]\[ \boxed{11.85} \][/tex]
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