Find the area enclosed between the curve y=4-x²

To find the area enclosed between the curve y = 4 - x², we need to integrate the equation within the appropriate bounds.

First, let's find the x-values where the curve intersects the x-axis by setting y = 0:

0 = 4 - x²

Rearranging the equation, we have:

x² = 4

Taking the square root of both sides, we get:

x = ±√4

Therefore, the curve intersects the x-axis at x = -2 and x = 2.

To calculate the area enclosed between the curve and the x-axis, we integrate the equation y = 4 - x² from x = -2 to x = 2:

Area = ∫(from -2 to 2) (4 - x²) dx

Using the power rule of integration, we can rewrite the integral:

Area = ∫(from -2 to 2) (4 - x²) dx = [4x - (1/3)x³] (from -2 to 2)

Now, substitute the upper and lower bounds into the equation:

Area = [4(2) - (1/3)(2)³] - [4(-2) - (1/3)(-2)³] = [8 - (8/3)] - [-8 - (-8/3)] = [24/3 - 8/3] - [-24/3 + 8/3] = 16/3 + 16/3 = 32/3

Therefore, the area enclosed between the curve y = 4 - x² and the x-axis is 32/3 square units.