To find the area between two curves, we will use the concept of definite integrals.
Given two functions, y = 4x^2 - 4 and y = -5x^2 + 5, we can find the area between these two curves from x = -1 to x = 1 by finding the definite integral of the absolute value of their difference over the interval from -1 to 1.
First, subtract y = -5x^2 + 5 from y = 4x^2 - 4, yielding 9x^2 - 9.
Now, since we are finding the area, we need the absolute value of this function. This is necessary because the area cannot be negative.
The function is symmetric around the y-axis, so we can integrate from 0 to 1 and multiply by 2 to get the result from -1 to 1.
Now the problem is simplified to finding
2 * ∫ from 0 to 1 |9x^2 - 9| dx.
The absolute value function will change the sign of the values that make 9x^2 - 9 negative.
Since we are working in the interval from 0 to 1, 9x^2 - 9 is always negative. Thus, we remove the absolute value bars and change the sign of the function inside to get
2 * ∫ from 0 to 1 (9 - 9x^2) dx.
Now we need to calculate a definite integral.
∫ from 0 to 1 (9 - 9x^2) dx = [9x - 3x^3] from 0 to 1 = 9*1 - 3*1^3 - (9*0 - 3*0^3) = 6.
Remember, this is only half of the solution because we halved the interval in order to get rid of the absolute value.
So, multiplying 6 by 2, we get a final solution of 12.
Therefore, the area of the region between the curves y = 4x^2 - 4 and y = -5x^2 + 5 from x = -1 to x = 1 is 12 square units.