The area of the curve $y^{2} = (7 - x) (5 + x)$ above x-axis and between the ordinates $x = - 5$ and $x = 1$ is $n π$ . Find $n$ .

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Solution

$y^{2} = 36 - (x - 1)^{2}$
$(x - 1)^{2} + y^{2} = 36$
$\int_{- 5}^{1} \sqrt{36 - (x - 1)^{2}} d x$
$\Rightarrow$ $[3 \sqrt{1 - \frac{(x - 1)^{2}}{36}} (x - 1) + 18 {sin}^{- 1} (\frac{x - 1}{6})]_{- 5}^{1}$
$\Rightarrow$ $- 18 {sin}^{- 1} (\frac{- 5 - 1}{6}) = - 18 {sin}^{- 1} (- 1) = - 18 \times (\frac{- π}{2})$
$\Rightarrow$ $9 π$

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