To find the volume enclosed by the parabolic cylinder and the given planes, we need to subtract the volume under the parabolic cylinder between the two planes from the volume under the upper plane between the same limits.
First, let's find the limits of integration. Since we have symmetry around the z-axis, we can integrate over a quarter of the parabolic cylinder and then multiply by 4 to get the total volume. Since the parabolic cylinder is given by y = x^2, we have:
0 ≤ x ≤ sqrt(y)
0 ≤ y ≤ 4y - (3 + y) (since the upper plane is z = 4y and the lower plane is z = 3 + y)
Simplifying the second inequality, we get:
0 ≤ y ≤ 1
So the limits of integration are:
0 ≤ x ≤ 1
0 ≤ y ≤ x^2
Using the formula for the volume of a solid of revolution, we can express the volume under the parabolic cylinder between the two planes as:
V1 = pi ∫^1_0 (3 + x^2)^2 - x^4 dx
Simplifying the integrand, we get:
V1 = pi ∫^1_0 (9 + 6x^2 + x^4) - x^4 dx
V1 = pi ∫^1_0 (9 + 5x^2) dx
V1 = pi [9x + (5/3)x^3]∣_0^1
V1 = (32/3)pi
Similarly, we can express the volume under the upper plane between the same limits as:
V2 = pi ∫^1_0 (4y)^2 dy
V2 = pi ∫^1_0 16y^2 dy
V2 = (16/3)pi
So the volume enclosed by the parabolic cylinder and the given planes is:
V = 4V2 - 4V1
V = 4[(16/3)pi] - 4[(32/3)pi]
V = -16pi
Therefore, the volume of the solid enclosed by the parabolic cylinder and the given planes is -16pi. Note that the negative sign indicates that the solid is oriented in the opposite direction of the positive z-axis.