Find the area enclosed between the curve y=x²-1 and y=7-x

To find the area enclosed between two curves, you need to first find the points of intersection.

Setting the two equations equal to each other, we have:

x² - 1 = 7 - x

Rearranging the equation:

x² + x - 8 = 0

Factoring the quadratic equation:

(x + 4)(x - 2) = 0

This gives us two possible values for x: x = -4 or x = 2.

Now, we can set up the integral to find the area:

Area = ∫(7 - x) - (x² - 1) dx, for x = -4 to x = 2

Simplifying the integral:

Area = ∫(8 - 2x - x²) dx, for x = -4 to x = 2

Applying the integral:

Area = [8x - x²/2 - x³/3] from -4 to 2

Evaluating the integral at the upper and lower limits:

Area = [(8(2) - (2)²/2 - (2)³/3)] - [(8(-4) - (-4)²/2 - (-4)³/3)]

Area = [(16 - 2 - 8/3)] - [(-32 - 8 - 64/3)]

Area = [16 - 2 - 8/3] - [-32 - 8 - 64/3]

Area = 10/3 - (-104/3)

Area = 114/3

Therefore, the area enclosed between the curves y = x² - 1 and y = 7 - x is 114/3 square units.