correct answer is (drum roll) = 21 1/3
to avoid beating around the bush
but if you like to beat bushes, (George HW, H, Jeb, Pete, if you're a Democrat) here's the fewest to beat up:
integrate -x^2+ 10 -(x^2+2) with respect to x
= integral of -2x^2 +8 dx
= (-2/3)x^3 +8x + a constant you can ignore as it cancels out when you get the definite integral with limits of integration
evaluated between the two intersection points which are where the two parabolas intersect
where x^2+2 =-x^2+10
2x^2=8
x = sqr(8/2 = sqr4=+/-2 the x coordinate of the intersection points (2,-6) and (2,6)
=2[(-2/3)(2^3) +8(2)] = 2[-16/3 +16] = 2(2/3)16 = 32/3 x 2 = 64/3 = 21 1/3 square units
the area =the overlap of two parabolas, one upward opening, the other downward opening
above calculations used the axis of symmetry, the y axis, then calculated form 0 to 2, found that area, then doubled it, so no need to get bogged down in limits from -2 to 2 which get tedious quickly
