Verified answer

1) Look at a general sketch of both. You'll see that f(x) intersects g(x) at X = {-sqrt(2),sqrt(2)}. You can solve this numerically by setting f(x) = g(x):

4 - x^2 = x^2

4 = 2(x^2)

2 = x^2 --> x = +-sqrt(2)

So, now you know your two endpoints, because the points of intersection will form a closed area. Now, by looking at the graph, you'll see that f(x) is always on top of g(x) within your domain.

To solve for the area, simply integrate the difference of the functions over your domain... in english:

int(-sqrt(2),sqrt(2)) [f(x) - g(x)]

2) You might remember that 'int(A -B)' is the same as saying 'int(A)' - 'int(B)' ... so let's do that:

f(x) int(-sqrt(2),sqrt(2)) [f(x)] ---> [4x -(x^3)/3](-sqrt(2),sqrt(2))

g(x) int(-sqrt(2),sqrt(2)) [f(x)] ---> [(x^3)/3](-sqrt(2),sqrt(2))

int[f(x)] = [(4*sqrt(2) - (sqrt(2)^3)/3] - [(4*-sqrt(2) - (-sqrt(2)^3)/3] = X

int[g(x)] = [(4*sqrt(2)] - [(4*-sqrt(2)] = Y

int(f(x) - g(x)) = X - Y

... you should be able to take it from there, hope that helps!