Final answer:
The volume of the solid S, which is the region enclosed by the parabola y = 4 - x², can be found by integrating the cross-sectional area A(x) of the solid from the lower limit to the upper limit. The lower limit is -2 and the upper limit is 2. The volume V is equal to 16.
Step-by-step explanation:
The volume V of the solid S can be found by integrating the cross-sectional area A(x) of the solid from the lower limit to the upper limit. The base of the solid is the region enclosed by the parabola y = 4 - x². To find A(x), we need to find the distance between the upper and lower curves of the region enclosed by the parabola. Setting the equations of the two curves equal to each other, we get:
4 - x² = 0
x² = 4
x = ±2
So, the lower limit is -2 and the upper limit is 2. Now, we can find A(x) by subtracting the equation of the lower curve from the equation of the upper curve:
A(x) = (4 - x²) - 0 = 4 - x²
Now, we can integrate A(x) with respect to x from -2 to 2 to get the volume V:
V = ∫[(4 - x²) dx] from -2 to 2
V = ∫[4 - x²] dx from -2 to 2
V = [4x - (x³/3)] from -2 to 2
Plugging in the limits, we get:
V = [4(2) - (2³/3)] - [4(-2) - ((-2)³/3)]
V = [8 - 8/3] - [-8 - 8/3]
V = (24/3 - 8/3) - (-24/3 - 8/3)
V = 16/3 - (-32/3)
V = 16/3 + 32/3
V = 48/3
V = 16