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

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(C) The given equation of the curve is $x + 2y + 8 = 0$,which can be rewritten as $x = -2y - 8$.The area $A$ of the region bounded by the curve $x = f

Vedclass · Accepted Answer

$8$

Vedclass · Answer

(C) The given equation of the curve is $x + 2y + 8 = 0$,which can be rewritten as $x = -2y - 8$.The area $A$ of the region bounded by the curve $x = f(y)$ and the lines $y = c$ and $y = d$ is given by the integral $A = \int_{c}^{d} |x| \, dy$.Here,$c = -3$ and $d = -1$.So,$A = \int_{-3}^{-1} |-2y - 8| \, dy$.Since for $y \in [-3, -1]$,the value of $-2y - 8$ is negative (e.g.,at $y = -2$,$-2(-2) - 8 = 4 - 8 = -4$),we have $|-2y - 8| = -(-2y - 8) = 2y + 8$.Thus,$A = \int_{-3}^{-1} (2y + 8) \, dy$.Integrating,we get $A = [y^2 + 8y]_{-3}^{-1}$.Evaluating at the limits: $A = ((-1)^2 + 8(-1)) - ((-3)^2 + 8(-3))$.$A = (1 - 8) - (9 - 24) = -7 - (-15) = -7 + 15 = 8$.Therefore,the area is $8$ sq. units.

The area of the region bounded by the curve $x + 2y + 8 = 0$ and the lines $y = -3$ and $y = -1$ is . . . . . . sq. units.

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