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Sagot :
To estimate the value of the definite integral [tex]\(\int_1^9 4x^3 \, dx\)[/tex] using a Riemann Sum with an overestimation for [tex]\( n = 4 \)[/tex] subdivisions, follow these steps:
1. Determine the width of each subinterval (Δx):
[tex]\[ \Delta x = \frac{b - a}{n} = \frac{9 - 1}{4} = \frac{8}{4} = 2 \][/tex]
2. Define the function to be integrated [tex]\( f(x) = 4x^3 \)[/tex].
3. Determine the sample points, specifically the right endpoints for the overestimate:
- The right endpoints are calculated as follows [tex]\( x_i = a + i\Delta x \)[/tex] for [tex]\( i = 1, 2, 3, 4 \)[/tex].
- Calculating the right endpoints:
[tex]\[ x_1 = 1 + 1 \cdot 2 = 3 \][/tex]
[tex]\[ x_2 = 1 + 2 \cdot 2 = 5 \][/tex]
[tex]\[ x_3 = 1 + 3 \cdot 2 = 7 \][/tex]
[tex]\[ x_4 = 1 + 4 \cdot 2 = 9 \][/tex]
- Therefore, our right endpoints are [tex]\(3, 5, 7,\)[/tex] and [tex]\(9\)[/tex].
4. Evaluate the function at these right endpoints:
[tex]\[ f(3) = 4 \cdot 3^3 = 4 \cdot 27 = 108 \][/tex]
[tex]\[ f(5) = 4 \cdot 5^3 = 4 \cdot 125 = 500 \][/tex]
[tex]\[ f(7) = 4 \cdot 7^3 = 4 \cdot 343 = 1372 \][/tex]
[tex]\[ f(9) = 4 \cdot 9^3 = 4 \cdot 729 = 2916 \][/tex]
5. Calculate the Riemann sum (overestimate):
- Multiply each function value at the right endpoint by the width of the subinterval [tex]\( \Delta x \)[/tex]:
[tex]\[ \text{Riemann Sum} = \Delta x \left[f(3) + f(5) + f(7) + f(9)\right] \][/tex]
- Substitute the values:
[tex]\[ \text{Riemann Sum} = 2 \left[108 + 500 + 1372 + 2916\right] \][/tex]
- Now sum the values inside the brackets:
[tex]\[ 108 + 500 + 1372 + 2916 = 4896 \][/tex]
- Multiply by [tex]\( \Delta x \)[/tex]:
[tex]\[ 2 \times 4896 = 9792 \][/tex]
Therefore, the estimate of the integral using a Riemann Sum with [tex]\( n = 4 \)[/tex] subdivisions, which is an overestimate, is:
[tex]\[ \boxed{9792} \][/tex]
1. Determine the width of each subinterval (Δx):
[tex]\[ \Delta x = \frac{b - a}{n} = \frac{9 - 1}{4} = \frac{8}{4} = 2 \][/tex]
2. Define the function to be integrated [tex]\( f(x) = 4x^3 \)[/tex].
3. Determine the sample points, specifically the right endpoints for the overestimate:
- The right endpoints are calculated as follows [tex]\( x_i = a + i\Delta x \)[/tex] for [tex]\( i = 1, 2, 3, 4 \)[/tex].
- Calculating the right endpoints:
[tex]\[ x_1 = 1 + 1 \cdot 2 = 3 \][/tex]
[tex]\[ x_2 = 1 + 2 \cdot 2 = 5 \][/tex]
[tex]\[ x_3 = 1 + 3 \cdot 2 = 7 \][/tex]
[tex]\[ x_4 = 1 + 4 \cdot 2 = 9 \][/tex]
- Therefore, our right endpoints are [tex]\(3, 5, 7,\)[/tex] and [tex]\(9\)[/tex].
4. Evaluate the function at these right endpoints:
[tex]\[ f(3) = 4 \cdot 3^3 = 4 \cdot 27 = 108 \][/tex]
[tex]\[ f(5) = 4 \cdot 5^3 = 4 \cdot 125 = 500 \][/tex]
[tex]\[ f(7) = 4 \cdot 7^3 = 4 \cdot 343 = 1372 \][/tex]
[tex]\[ f(9) = 4 \cdot 9^3 = 4 \cdot 729 = 2916 \][/tex]
5. Calculate the Riemann sum (overestimate):
- Multiply each function value at the right endpoint by the width of the subinterval [tex]\( \Delta x \)[/tex]:
[tex]\[ \text{Riemann Sum} = \Delta x \left[f(3) + f(5) + f(7) + f(9)\right] \][/tex]
- Substitute the values:
[tex]\[ \text{Riemann Sum} = 2 \left[108 + 500 + 1372 + 2916\right] \][/tex]
- Now sum the values inside the brackets:
[tex]\[ 108 + 500 + 1372 + 2916 = 4896 \][/tex]
- Multiply by [tex]\( \Delta x \)[/tex]:
[tex]\[ 2 \times 4896 = 9792 \][/tex]
Therefore, the estimate of the integral using a Riemann Sum with [tex]\( n = 4 \)[/tex] subdivisions, which is an overestimate, is:
[tex]\[ \boxed{9792} \][/tex]
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