Set up the triple integral for the volume of the solid that is the common interior below the sphere x2+ y2+ z2= 8 and above the paraboloid z= 1/2 (x2+ y2)

Step-by-step explanation:

The common region enclosed above the paraboloid and below the sphere is the solid.

Let's find the limit bounds for .

The region is bounded above by the sphere .

Isolate from the equation above.

On the other hand, the region is bounded below by the paraboloid  

Therefore, we obtain

Now, we need to find the intersection of the sphere and the paraboloid. To do so, we need to solve the following system of equations

.

Substitute  the second equation in the first equation. We obtain

Hence, the paraboloid and the sphere intersect when .

Substituting for in the equation above gives

which is a circle with a radius .

Now, we can find the bond for and .

For , we obtain

.

For , we have

.

Therefore, the needed triple integral is

⊂⊃⊂⊃ω

step-by-step explanation:

i hope this : )