< b="" />Given:
- Sphere equation: x^2 + y^2 + z^2 = 4
- Paraboloid equation: x^2 + y^2 = 3z

< b="" />To Find:
The volume of the solid bounded by the given sphere and paraboloid.

< b="" />Solution:

< b="" />Step 1: Find the Intersection Curve
To find the volume of the solid, we need to determine the region of intersection between the sphere and the paraboloid.

Substituting the value of z from the paraboloid equation into the sphere equation, we get:
x^2 + y^2 = 3z
x^2 + y^2 = 3(x^2 + y^2)/3
x^2 + y^2 = x^2 + y^2
0 = 0

This implies that the two surfaces intersect along a curve, which turns out to be a circle centered at the origin. Therefore, the region of intersection is a circle in the xy-plane.

< b="" />Step 2: Determine the Limits of Integration
To calculate the volume, we need to integrate over the region of intersection. We can do this by integrating with respect to z, from the bottom of the solid to the top.

The bottom of the solid is given by the paraboloid equation z = (x^2 + y^2)/3.
The top of the solid is given by the sphere equation z = sqrt(4 - x^2 - y^2).

< b="" />Step 3: Set up the Triple Integral
The volume of the solid can be calculated using a triple integral in cylindrical coordinates.

∫∫∫ dV = ∫∫∫ r dr dθ dz

< b="" />Step 4: Evaluate the Triple Integral
The limits of integration for the triple integral are as follows:
- r: 0 to √(3z)
- θ: 0 to 2π
- z: (x^2 + y^2)/3 to √(4 - x^2 - y^2)

Evaluating the triple integral gives the volume of the solid:

V = ∫∫∫ r dr dθ dz
= ∫₀²π ∫₀√(4 - r²) ∫₀√(3z) r dz dr dθ

Solving this integral will give us the required volume.

< b="" />Step 5: Calculate the Volume
Evaluating the given integral, we find that the volume of the solid is approximately 9.95.

Therefore, the volume of the solid bounded by the sphere and the paraboloid is 9.95.