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To find the area of the region between the graph of the function [tex]\( f(x) = \frac{5}{7} x^{\frac{2}{7}} + \frac{1}{2} x^{\frac{4}{9}} + 6 \)[/tex] and the [tex]\( x \)[/tex]-axis on the interval [tex]\([4, 8]\)[/tex], we use the Fundamental Theorem of Calculus. The steps are as follows:
1. Define the Integral:
We need to compute the definite integral of the function [tex]\( f(x) \)[/tex] from 4 to 8:
[tex]\[ \int_{4}^{8} \left( \frac{5}{7} x^{\frac{2}{7}} + \frac{1}{2} x^{\frac{4}{9}} + 6 \right) \, dx \][/tex]
2. Compute the Antiderivative:
First, we find the antiderivative [tex]\( F(x) \)[/tex] of the function [tex]\( f(x) \)[/tex]:
[tex]\[ \int \left( \frac{5}{7} x^{\frac{2}{7}} + \frac{1}{2} x^{\frac{4}{9}} + 6 \right) \, dx \][/tex]
The antiderivatives of the individual terms are:
- For [tex]\( \frac{5}{7} x^{\frac{2}{7}} \)[/tex]:
[tex]\[ \int \frac{5}{7} x^{\frac{2}{7}} \, dx = \frac{5}{7} \cdot \frac{7}{9} x^{\frac{9}{7}} = \frac{5}{9} x^{\frac{9}{7}} \][/tex]
- For [tex]\( \frac{1}{2} x^{\frac{4}{9}} \)[/tex]:
[tex]\[ \int \frac{1}{2} x^{\frac{4}{9}} \, dx = \frac{1}{2} \cdot \frac{9}{13} x^{\frac{13}{9}} = \frac{9}{26} x^{\frac{13}{9}} \][/tex]
- For 6:
[tex]\[ \int 6 \, dx = 6x \][/tex]
Therefore, the antiderivative [tex]\( F(x) \)[/tex] is:
[tex]\[ F(x) = \frac{5}{9} x^{\frac{9}{7}} + \frac{9}{26} x^{\frac{13}{9}} + 6x + C \][/tex]
where [tex]\( C \)[/tex] is the constant of integration, which is not needed for definite integrals.
3. Evaluate the Definite Integral:
We now evaluate [tex]\( F(x) \)[/tex] at the upper and lower limits of the interval and subtract:
[tex]\[ \int_{4}^{8} f(x) \, dx = F(8) - F(4) \][/tex]
Substituting the values of 8 and 4 into [tex]\( F(x) \)[/tex]:
[tex]\[ F(8) = \frac{5}{9} \cdot 8^{\frac{9}{7}} + \frac{9}{26} \cdot 8^{\frac{13}{9}} + 6 \cdot 8 \][/tex]
[tex]\[ F(4) = \frac{5}{9} \cdot 4^{\frac{9}{7}} + \frac{9}{26} \cdot 4^{\frac{13}{9}} + 6 \cdot 4 \][/tex]
Now, calculate [tex]\( F(8) \)[/tex] and [tex]\( F(4) \)[/tex]:
[tex]\[ F(8) \approx 90.780 \][/tex]
[tex]\[ F(4) \approx 57.617 \][/tex]
Therefore, the definite integral is:
[tex]\[ \int_{4}^{8} f(x) \, dx = 90.780 - 57.617 \approx 33.163 \][/tex]
4. Round the Result:
The area of the region, rounded to the nearest integer, is:
[tex]\[ \boxed{33 \text{ units}^2} \][/tex]
Hence, the correct answer is [tex]\( 33 \text{ units}^2 \)[/tex].
1. Define the Integral:
We need to compute the definite integral of the function [tex]\( f(x) \)[/tex] from 4 to 8:
[tex]\[ \int_{4}^{8} \left( \frac{5}{7} x^{\frac{2}{7}} + \frac{1}{2} x^{\frac{4}{9}} + 6 \right) \, dx \][/tex]
2. Compute the Antiderivative:
First, we find the antiderivative [tex]\( F(x) \)[/tex] of the function [tex]\( f(x) \)[/tex]:
[tex]\[ \int \left( \frac{5}{7} x^{\frac{2}{7}} + \frac{1}{2} x^{\frac{4}{9}} + 6 \right) \, dx \][/tex]
The antiderivatives of the individual terms are:
- For [tex]\( \frac{5}{7} x^{\frac{2}{7}} \)[/tex]:
[tex]\[ \int \frac{5}{7} x^{\frac{2}{7}} \, dx = \frac{5}{7} \cdot \frac{7}{9} x^{\frac{9}{7}} = \frac{5}{9} x^{\frac{9}{7}} \][/tex]
- For [tex]\( \frac{1}{2} x^{\frac{4}{9}} \)[/tex]:
[tex]\[ \int \frac{1}{2} x^{\frac{4}{9}} \, dx = \frac{1}{2} \cdot \frac{9}{13} x^{\frac{13}{9}} = \frac{9}{26} x^{\frac{13}{9}} \][/tex]
- For 6:
[tex]\[ \int 6 \, dx = 6x \][/tex]
Therefore, the antiderivative [tex]\( F(x) \)[/tex] is:
[tex]\[ F(x) = \frac{5}{9} x^{\frac{9}{7}} + \frac{9}{26} x^{\frac{13}{9}} + 6x + C \][/tex]
where [tex]\( C \)[/tex] is the constant of integration, which is not needed for definite integrals.
3. Evaluate the Definite Integral:
We now evaluate [tex]\( F(x) \)[/tex] at the upper and lower limits of the interval and subtract:
[tex]\[ \int_{4}^{8} f(x) \, dx = F(8) - F(4) \][/tex]
Substituting the values of 8 and 4 into [tex]\( F(x) \)[/tex]:
[tex]\[ F(8) = \frac{5}{9} \cdot 8^{\frac{9}{7}} + \frac{9}{26} \cdot 8^{\frac{13}{9}} + 6 \cdot 8 \][/tex]
[tex]\[ F(4) = \frac{5}{9} \cdot 4^{\frac{9}{7}} + \frac{9}{26} \cdot 4^{\frac{13}{9}} + 6 \cdot 4 \][/tex]
Now, calculate [tex]\( F(8) \)[/tex] and [tex]\( F(4) \)[/tex]:
[tex]\[ F(8) \approx 90.780 \][/tex]
[tex]\[ F(4) \approx 57.617 \][/tex]
Therefore, the definite integral is:
[tex]\[ \int_{4}^{8} f(x) \, dx = 90.780 - 57.617 \approx 33.163 \][/tex]
4. Round the Result:
The area of the region, rounded to the nearest integer, is:
[tex]\[ \boxed{33 \text{ units}^2} \][/tex]
Hence, the correct answer is [tex]\( 33 \text{ units}^2 \)[/tex].
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