Triple integral to find Volume of Solid

[yyh1]

Homework Statement

Use a triple integral to find the volume of solid enclosed between the sphere and paraboloid.

Homework Equations

Equation for sphere x²+y²+z²=2a²
Equation for paraboloid az = x²+y² (a>0)

The Attempt at a Solution

Trying to find limits of integration:
For integration of dz, rearranging the both equation in terms of z, the limits are from
z= 1/a (x²+y²) to
z= SQRT (2a² - x² - y²)

Next i suppose i should find the equations in xy plane by solving the given equations simultaneously to determine where the sphere and paraboloid intersect. When i equate both equation, i got this expression
2a⁴= (x²+y²)²+a²(x²+y²). I got stuck here as I do not know how to find the limits for dx and dy. Do i need to use polar coordinates or sphere coordinates? Can anyone explain these to me? Thanks

 

[mighty2000]

!



To solve this problem using a triple integral, you will need to convert the given equations into spherical coordinates. This will make the integration process easier. The equations for sphere and paraboloid in spherical coordinates are:

Equation for sphere: ρ^2 = 2a^2
Equation for paraboloid: aρ cos(φ) = ρ^2 sin(φ)

To find the limits of integration, you will need to determine the points of intersection between the sphere and paraboloid. You can do this by setting the equations equal to each other and solving for ρ and φ. Once you have these points, you can determine the limits for ρ and φ.

For ρ, the limits will be from 0 to the distance between the origin and the point of intersection. For φ, the limits will be from 0 to the angle between the z-axis and the line connecting the origin and the point of intersection.

Once you have determined the limits for ρ and φ, you can integrate with respect to ρ, φ, and θ (from 0 to 2π) to find the volume of the solid enclosed between the sphere and paraboloid.

I hope this helps. Good luck with your problem!