Finding the volume of an object with 3 parametersI know how to find the volume of a sphere/ball around x-axis using: V = π ∫ a b f 2 ( x ) d xLets say if: x 2 + y 2 = r 2 We do: y = r 2 − x 2 So now: V = π ∫ − r r ( r 2 − x 2 ) 2 d xBut the problem starts here:Im given an equation of ellipsoid, it has 3 parameters: x 2 + 4 y 2 + 4 z 2 ≤ 4What do i do with the z parameter? How do i build y now? how does it fit to the equation by integrals of V?I would like an explanation and not just a solution - because its homework.

Question

Finding the volume of an object with 3 parametersI know how to find the volume of a sphere/ball around x-axis using:  V  =  π      ∫          a              b            f    2    (  x  )  d  xLets say if:      x    2    +      y    2    =      r    2  We do:   y  =            r      2        −          x      2      So now:   V  =  π      ∫          −      r              r        (            r      2        −          x      2            )    2    d  xBut the problem starts here:Im given an equation of ellipsoid, it has 3 parameters:      x    2    +  4      y    2    +  4      z    2    ≤  4What do i do with the z parameter? How do i build y now? how does it fit to the equation by integrals of V?I would like an explanation and not just a solution - because its homework.

ticotaku86 · Accepted Answer

Step 1In finding the volume of a ball       x    2    +      y    2    +      z    2    ≤      r    2  , you revolve the curve   y  =            r      2        −          x      2       about the y-axis. Notice that   y  =            r      2        −          x      2       is the equation you get by setting   z  =  0 in the equation for the sphere.You can do the same thing with the ellipsoid: set   z  =  0 to get the equation for the boundary of the rugby ball in the (x,y)-plane, and solve for y.      x    2    +  4      y    2    =  4    ⟹    y  =            4      −              x        2              2  Step 2Revolving this curve about the y-axis gives the volume,  π      ∫          −      2        2              (                        4          −                      x            2                          2            )        2          d    x  =      π    4        ∫          −      2        2    4  −      x    2          d    x  =            8      π        3

Finding the volume of an object with 3 parameters. given an equation of ellipsoid, it has 3 parameters: x^2+4y^2+4z^2 leq 4

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