definite intergration!!

First, the points of intersection between the curves are:

x2 - 6 = x

x2 - x - 6 = 0

(x - 3)(x + 2) = 0

x = 3 or -2

Also the x-intercepts of y = x2 - 6 are x = -√6 and x = √6

So the shaded area is given by:

∫(x = -√6 → -2) (6 - x2) dx + ∫(x = -2 → 0) (- x) dx + ∫(x = 0 → √6) x dx + ∫(x = √6 → 3) [x - (x2 - 6)] dx

= ∫(x = -√6 → -2) (6 - x2) dx + ∫(x = -2 → 0) (- x) dx + ∫(x = 0 → √6) x dx + ∫(x = √6 → 3) (6 + x - x2) dx

= [6x - x3/3] (x = -√6 → -2) + [-x2/2] (x = -2 → 0) + [x2/2] (x = 0 → √6) + [6x + x2/2 - x3/3] (x = √6 → 3)

= [(-12 + 8/3) - (-6√6 + 2√6)] + 2 + 3 + [(18 + 9/2 - 9) - (6√6 + 3 - 2√6)]

= (-28/3 + 4√6) + 5 + (27/2 - 3 - 4√6)

= -28/3 + 4√6 + 5 + 21/2 - 4√6

= 37/6