Problem:
Find the volume of the region bounded by the surfaces y = x^2, x = y^2, and the planes z = 0 and z = 3.

Solution:
To find the volume of the region bounded by the given surfaces, we need to set up the triple integral in the appropriate coordinate system. Let's consider using the cylindrical coordinate system since the given surfaces have a rotational symmetry.

Step 1: Identifying the Limits of Integration
We need to determine the limits of integration for each variable: ρ, φ, and z.

Limit of ρ:
Since the region is bounded by the surfaces y = x^2 and x = y^2, we can express ρ in terms of φ as follows:
ρ = f(φ) = cos(φ)^2.

Limit of φ:
To determine the limits of φ, we need to find the intersection points between the curves y = x^2 and x = y^2. Setting these equations equal to each other, we get x^2 = (x^2)^2, which simplifies to x^4 - x^2 = 0. Factoring out x^2, we have x^2(x^2 - 1) = 0. This gives us two possible values for x: x = 0 and x = ±1.

For x = 0, we have y = 0. Therefore, one intersection point is (0, 0).

For x = 1, we have y = 1. Therefore, another intersection point is (1, 1).

Hence, the limits of φ are 0 and π/4 since the curve y = x^2 lies in the first quadrant.

Limit of z:
The planes z = 0 and z = 3 define the limits of z as 0 and 3, respectively.

Step 2: Setting Up the Triple Integral
The volume of the region can be calculated using the triple integral:
V = ∫∫∫ dV,

where dV is the differential volume element in cylindrical coordinates given by dV = ρ dz dρ dφ.

Step 3: Evaluating the Triple Integral
Using the limits of integration determined in Step 1, the triple integral can be set up as follows:

V = ∫[0 to 3] ∫[0 to π/4] ∫[0 to cos(φ)^2] ρ dρ dφ dz.

Evaluating the innermost integral with respect to ρ, we get:

V = ∫[0 to 3] ∫[0 to π/4] (ρ^2/2) |[0 to cos(φ)^2] dφ dz.

Simplifying further, we have:

V = ∫[0 to 3] ∫[0 to π/4] (cos(φ)^6/2) dφ dz.

Evaluating the remaining integral with respect to φ, we get:

V = ∫[0 to 3] (1/7)(3(2 - 7cos(π/4)^6)) dz.

Simplifying further, we have:

V = (1/7)(2 -