![V = 64\pi . [\frac{16}{3} - \sqrt{3} ]](/tpl/images/0709/2347/8f760.png)
 unit^3 .. ( or equivalent )
Step-by-step explanation:
Solution:-
- The following surfaces are given as follows:
                
- We will first have to investigate the region that lies inside both the cylinder and the ellipsoid or the region common to both surface.
Step 1: Coordinate transformation ( Cartesian ( x,y,z ) -> Polar ( r,θ,z )
- To convert the cartesian coordinates to polar/cylindrical coordinate system we will take the help of conversion equation given below:
                
- Make the substitution of the above equation in the given equation of the cylinder as follows:
                
- Make the substitution of the transformation equation into the ellipsoid equation as follows:
               
Step 2: Sketch/Plot the region of volume
- We can either sketch or plot the surfaces in the cartesian coordinate system. I have utilized " Geogebra " 3D graphing utility.
- The purpose of graphing/sketching the surfaces is to "visualize" the bounds of the volume that lies inside both of the surfaces. This will help us in step 3 to set-up limits of integration. Moreover, the sketches also help us to see whether the enclosed volume is symmetrical about any axis.
- The plot is given as an attachment. From the plot we see that the volume of integration lies both above and below x-y plane ( z = 0 ). This result can be seen from the polar equation of surfaces ( +/- ) obtained.
- The Volume is defined by a shape of a vessel. I.e " Cylinder with two hemispherical caps on the circular bases "
- We see that enclosed volume is symmetrical about ( x-y ) plane. Each section of volume ( above and below x-y plane ) is bounded by the planes:
             Upper half Volume:
             
            Lower Half Volume:
             
- We will simplify our integral manipulation by using the above symmetry and consider the upper half volume and multiply the result by 2.
Note: We can also find quadrant symmetry of the volume defined by the circular projection of the volume on ( x-y plane ); however, the attempt would lead to higher number of computations/tedious calculations. This invalidates the purpose of using symmetry to simplify mathematical manipulations. Â
Step 3: Set-up triple integral in the cylindrical coordinate system
- The general formulation for setting up triple integrals in cylindrical coordinate system depends on the order of integration.
- We will choose the following order in the direction of "easing symmetry" i.e: (dz.dr.dθ) Â
               
- Define the limits:
         a: z = 0 - > ( x-y plane, symmetry plane )
         b: z = 2√(16 - r^2) - > ( upper surface of ellipsoid )
      * Multiply the integral ( dz ) by " x2 "
         c: r = 0  ( symmetry axis )
         d: r = +2 ( circle of radius 2 units - higher )
       * Multiply the integral ( r.dr ) by " x2 "
         e: θ = 0 ( initial point of angle sweep )
         f: θ = 2π ( final point of angle sweep )
- The integral formulation becomes:
            
Step 4: Integral evaluation
- The last step is to perform the integration of the formulation derived in previous step.
         Â
            ![V = 4 \int\limits^f_e \int\limits^d_c{2.r.\sqrt{16 - r^2} } \, .dr.dQ\\\\V = 4 \int\limits^f_e {-\frac{2}{3} . (16 - r^2)^\frac{3}{2} } \,|\Limits^2_0 . dQ\\\\V = -\frac{8}{3} \int\limits^f_e { [ 24\sqrt{3} - 64] } .dQ\\\\V = -\frac{8}{3}. [ 24\sqrt{3} - 64] . 2\pi \\\\V = 64\pi . [\frac{16}{3} - \sqrt{3} ]](/tpl/images/0709/2347/bf7ff.png)
Answer.
