Question 1172398
Let's find the volume of the solid.

**1. Determine the Region in the xy-Plane**

* We need to find the intersection of the parabolas y = x^2 and y = 2 - x^2.
* Set them equal: x^2 = 2 - x^2
* 2x^2 = 2
* x^2 = 1
* x = ±1
* When x = ±1, y = 1.
* The region in the xy-plane is bounded by these parabolas, with x ranging from -1 to 1.

**2. Set up the Triple Integral**

* The volume is given by the triple integral:
    * V = ∫∫∫ dV
* The limits of integration are:
    * z: 0 to y + 3
    * y: x^2 to 2 - x^2
    * x: -1 to 1

* The integral becomes:
    * V = ∫(from -1 to 1) ∫(from x^2 to 2 - x^2) ∫(from 0 to y + 3) dz dy dx

**3. Evaluate the Integral**

* First, integrate with respect to z:
    * ∫(from 0 to y + 3) dz = [z](from 0 to y + 3) = y + 3
* Now, integrate with respect to y:
    * ∫(from x^2 to 2 - x^2) (y + 3) dy = [y^2/2 + 3y](from x^2 to 2 - x^2)
    * = [(2 - x^2)^2/2 + 3(2 - x^2)] - [(x^2)^2/2 + 3(x^2)]
    * = [4 - 4x^2 + x^4]/2 + 6 - 3x^2 - x^4/2 - 3x^2
    * = 2 - 2x^2 + x^4/2 + 6 - 3x^2 - x^4/2 - 3x^2
    * = 8 - 8x^2

* Finally, integrate with respect to x:
    * ∫(from -1 to 1) (8 - 8x^2) dx = [8x - 8x^3/3](from -1 to 1)
    * = [8(1) - 8(1)^3/3] - [8(-1) - 8(-1)^3/3]
    * = [8 - 8/3] - [-8 + 8/3]
    * = 8 - 8/3 + 8 - 8/3
    * = 16 - 16/3
    * = (48 - 16)/3
    * = 32/3

**Answer**

The volume of the solid is 32/3 cubic units.