find the area of the region bounded by the curves y=x^2-1 and y =cos(x)

Answer 1

the hardest part is finding the intersection of the two curves,
x^2 - 1 = cosx
I will let Wolfram do it:
http://www.wolframalpha.com/input/?i=x%5E2+-+1+%3D+cosx

Because of the symmetry we could go from 0 to 1.1765 and double the answer

area = 2 ∫(cosx - x^2 + 1)dx from 0 to 1.1765
= 2[sinx - (1/3)x^3 + x] from 0 to 1.1765
= 2(.923237... - .542818... + 1.1765 - (0 - 0 + 0))
= appr 5.285