The area bounded by the curve x = $∣ y^{2} - 1 ∣$ and the line y = x -5 is

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Solution

$\begin{matrix} S o l v i n g b o t h e q u a t i o n s, x = y + 5 x = y^{2} - 1 y + 5 = y^{2} - 1 y^{2} - y - 6 = 0 y^{2} - 3 y + 2 y - 6 = 0 y (y - 3) + 2 (y - 3) x = (3, 8) y = (- 2, 3) A r e a = \int_{3}^{8} ((x - 5) - \sqrt{x + 1}) d x =_{3}^{8} [\frac{x^{2}}{2} - 5 x - \frac{2 {(x + 1)}^{\frac{3}{2}}}{3} = \frac{55}{2} - 25 - \frac{2 (27 - 8)}{3} = ∣ ∣ \frac{55}{2} - 25 - 2 \times \frac{19}{3} ∣ ∣ = \frac{61}{6} s q u a r e u n i t s . \end{matrix}$

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