The area of the region bounded by the curve x | vedclass

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(C) Given the equation of the curve $x+2y=8$,we can express $y$ in terms of $x$ as: $2y = 8 - x$ $y = \frac{8-x}{2} = 4 - \frac{x}{2}$ To find the are

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$10$

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(C) Given the equation of the curve $x+2y=8$,we can express $y$ in terms of $x$ as: $2y = 8 - x$ $y = \frac{8-x}{2} = 4 - \frac{x}{2}$ To find the area bounded by the curve,the $X$-axis,and the lines $x=1$ and $x=5$,we use the definite integral: $	ext{Area} = \int_{1}^{5} y \, dx$ $	ext{Area} = \int_{1}^{5} (4 - \frac{x}{2}) \, dx$ Integrating term by term: $	ext{Area} = [4x - \frac{x^2}{4}]_{1}^{5}$ Evaluating at the limits: $	ext{Area} = (4(5) - \frac{5^2}{4}) - (4(1) - \frac{1^2}{4})$ $	ext{Area} = (20 - \frac{25}{4}) - (4 - \frac{1}{4})$ $	ext{Area} = 20 - 6.25 - 4 + 0.25$ $	ext{Area} = 16 - 6 = 10$ Thus,the area is $10$ sq. units. The correct option is $C$.

The area of the region bounded by the curve $x+2y=8$,the $X$-axis,and the lines $x=1$ and $x=5$ using integration is . . . . . . sq. units.

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