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Sagot :
To approximate the integral [tex]\(\int_8^{12} 5 \sqrt{x-2} \, dx\)[/tex], we will use a left Riemann sum with 4 subintervals.
Step-by-Step Solution:
1. Define the function and interval:
The function to be integrated is [tex]\( f(x) = 5 \sqrt{x - 2} \)[/tex].
The interval of integration is [tex]\([8, 12]\)[/tex].
2. Determine the subinterval width:
The interval [tex]\([8, 12]\)[/tex] is divided into 4 subintervals. Therefore, the width of each subinterval, [tex]\(\Delta x\)[/tex], is:
[tex]\[ \Delta x = \frac{12 - 8}{4} = 1 \][/tex]
3. Identify the left endpoints of each subinterval:
The left endpoints [tex]\( x_i \)[/tex] of each of the 4 subintervals are:
[tex]\[ x_0 = 8, \quad x_1 = 8 + 1 = 9, \quad x_2 = 9 + 1 = 10, \quad x_3 = 10 + 1 = 11 \][/tex]
4. Evaluate the function at each left endpoint:
[tex]\[ f(x_0) = 5 \sqrt{8 - 2} \][/tex]
[tex]\[ f(x_1) = 5 \sqrt{9 - 2} \][/tex]
[tex]\[ f(x_2) = 5 \sqrt{10 - 2} \][/tex]
[tex]\[ f(x_3) = 5 \sqrt{11 - 2} \][/tex]
These evaluate to:
[tex]\[ f(8) = 5 \sqrt{6} \approx 12.24744871391589 \][/tex]
[tex]\[ f(9) = 5 \sqrt{7} \approx 13.228756555322953 \][/tex]
[tex]\[ f(10) = 5 \sqrt{8} \approx 14.142135623730951 \][/tex]
[tex]\[ f(11) = 5 \sqrt{9} \approx 15.0 \][/tex]
5. Sum the function evaluations and multiply by [tex]\(\Delta x\)[/tex]:
[tex]\[ \text{Left Riemann Sum} = \Delta x \left( f(x_0) + f(x_1) + f(x_2) + f(x_3) \right) \][/tex]
[tex]\[ = 1 \left( 12.24744871391589 + 13.228756555322953 + 14.142135623730951 + 15.0 \right) \][/tex]
[tex]\[ \approx 1 \times 54.6183408929698 \][/tex]
[tex]\[ \approx 54.6183408929698 \][/tex]
Therefore, the left Riemann sum approximation for the integral [tex]\(\int_8^{12} 5 \sqrt{x-2} \, dx\)[/tex] using 4 subintervals is:
[tex]\[ \int_8^{12} 5 \sqrt{x-2} \, dx \approx 12.24744871391589 + 13.228756555322953 + 14.142135623730951 + 15.0 = 54.6183408929698 \][/tex]
Step-by-Step Solution:
1. Define the function and interval:
The function to be integrated is [tex]\( f(x) = 5 \sqrt{x - 2} \)[/tex].
The interval of integration is [tex]\([8, 12]\)[/tex].
2. Determine the subinterval width:
The interval [tex]\([8, 12]\)[/tex] is divided into 4 subintervals. Therefore, the width of each subinterval, [tex]\(\Delta x\)[/tex], is:
[tex]\[ \Delta x = \frac{12 - 8}{4} = 1 \][/tex]
3. Identify the left endpoints of each subinterval:
The left endpoints [tex]\( x_i \)[/tex] of each of the 4 subintervals are:
[tex]\[ x_0 = 8, \quad x_1 = 8 + 1 = 9, \quad x_2 = 9 + 1 = 10, \quad x_3 = 10 + 1 = 11 \][/tex]
4. Evaluate the function at each left endpoint:
[tex]\[ f(x_0) = 5 \sqrt{8 - 2} \][/tex]
[tex]\[ f(x_1) = 5 \sqrt{9 - 2} \][/tex]
[tex]\[ f(x_2) = 5 \sqrt{10 - 2} \][/tex]
[tex]\[ f(x_3) = 5 \sqrt{11 - 2} \][/tex]
These evaluate to:
[tex]\[ f(8) = 5 \sqrt{6} \approx 12.24744871391589 \][/tex]
[tex]\[ f(9) = 5 \sqrt{7} \approx 13.228756555322953 \][/tex]
[tex]\[ f(10) = 5 \sqrt{8} \approx 14.142135623730951 \][/tex]
[tex]\[ f(11) = 5 \sqrt{9} \approx 15.0 \][/tex]
5. Sum the function evaluations and multiply by [tex]\(\Delta x\)[/tex]:
[tex]\[ \text{Left Riemann Sum} = \Delta x \left( f(x_0) + f(x_1) + f(x_2) + f(x_3) \right) \][/tex]
[tex]\[ = 1 \left( 12.24744871391589 + 13.228756555322953 + 14.142135623730951 + 15.0 \right) \][/tex]
[tex]\[ \approx 1 \times 54.6183408929698 \][/tex]
[tex]\[ \approx 54.6183408929698 \][/tex]
Therefore, the left Riemann sum approximation for the integral [tex]\(\int_8^{12} 5 \sqrt{x-2} \, dx\)[/tex] using 4 subintervals is:
[tex]\[ \int_8^{12} 5 \sqrt{x-2} \, dx \approx 12.24744871391589 + 13.228756555322953 + 14.142135623730951 + 15.0 = 54.6183408929698 \][/tex]
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