Given:
Paraboloid equation: z = x^2 + y^2
Plane equation: z = 2y

To Find:
Volume of the solid bounded by the paraboloid and above the plane.

Solution:

Step 1: Graphical Representation
Let's first understand the shape of the paraboloid and the plane in order to visualize the solid bounded by them.

- The paraboloid z = x^2 + y^2 is a surface that opens upwards and extends infinitely.
- The plane z = 2y is a flat surface parallel to the x-y plane.

By plotting these two surfaces, we can see that the paraboloid intersects the plane at a certain height, forming a solid region above the plane.

Step 2: Setting up the Integral
To find the volume of the solid, we need to set up a triple integral over the region bounded by the paraboloid and the plane.

Let's consider the region R in the x-y plane that is bounded by the parabola x^2 + y^2 = 2y.

- We can rewrite this equation as x^2 + (y - 1)^2 = 1, which represents a circle with center (0, 1) and radius 1.
- This circle lies in the x-y plane.

Step 3: Integrating over the Region R
To set up the triple integral, we need to determine the limits of integration for x, y, and z.

- For x, we can use the limits of the circle in the x-y plane: -1 ≤ x ≤ 1.
- For y, the limits are given by the equation of the circle: 1 - √(1 - x^2) ≤ y ≤ 1 + √(1 - x^2).
- For z, the limits are given by the equations of the paraboloid and the plane: 2y ≤ z ≤ x^2 + y^2.

Step 4: Evaluating the Integral
Now, we can set up the triple integral and evaluate it to find the volume.

V = ∫∫∫ 1 dz dy dx over the region R

Using the limits of integration described in Step 3, we can evaluate the integral:

V = ∫(-1 to 1) ∫(1 - √(1 - x^2) to 1 + √(1 - x^2)) ∫(2y to x^2 + y^2) 1 dz dy dx

After performing the integration, the volume of the solid is found to be approximately 1.570.

Answer:
The volume of the solid bounded by the paraboloid z = x^2 + y^2 and above the plane z = 2y is 1.570.