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

To find the area enclosed between the curve y = 4 - x², we need to determine the boundaries of integration.

First, let's find where the curve intersects the x-axis by setting y to zero:

0 = 4 - x²

Rearranging the equation, we get:

x² = 4

Taking the square root of both sides, we have:

x = ±√4 = ±2

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

To find the area enclosed, we need to integrate the function y = 4 - x² with respect to x from x = -2 to x = 2.

∫[from -2 to 2] (4 - x²) dx

First, we can evaluate this indefinite integral:

∫ (4 - x²) dx = 4x - (1/3)x³ + C

Now, we can find the definite integral:

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

Plugging in the values, we get:

[4(2) - (1/3)(2)³] - [4(-2) - (1/3)(-2)³]

Simplifying further:

[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 is 32/3 square units.