How to calculate the volume of an ellipsoid with triple integralI'm having some troubles since this morning on an exercise. I need to find the volume of the ellipsoid defined by x 2 a 2 + y 2 a 2 + z 2 a 2 ≤ 1. So at the beginning I wrote { − a ≤ x ≤ a − b ≤ y ≤ b − c ≤ z ≤ c Then I wrote this as integral: ∫ − c c ∫ − b b ∫ − a a 1 d x d y d z.I found as a result : 8abcBut I knew it was incorrect. I browsed this website and many others for 2 hours and yeah I found that we need to express b and c in terms of x. But I don't know why.. Is there an intuitive way to understand that ? I have already done some double integral with b expressed in terms of a but I never figured out why..

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How to calculate the volume of an ellipsoid with triple integralI&#039;m having some troubles since this morning on an exercise. I need to find the volume of the ellipsoid defined by             x      2              a      2        +            y      2              a      2        +            z      2              a      2        ≤  1. So at the beginning I wrote       {                            −          a          ≤          x          ≤          a                                      −          b          ≤          y          ≤          b                                      −          c          ≤          z          ≤          c                        Then I wrote this as integral:       ∫          −      c              c            ∫          −      b              b            ∫          −      a              a        1  d  x  d  y  d  z.I found as a result : 8abcBut I knew it was incorrect. I browsed this website and many others for 2 hours and yeah I found that we need to express b and c in terms of x. But I don&#039;t know why.. Is there an intuitive way to understand that ? I have already done some double integral with b expressed in terms of a but I never figured out why..

kunstdansvo · Accepted Answer

Explanation:The easiest way to do this problem is to scale the axes by a, b, and c. That turns the ellipse into a sphere of radius 1 and multiplies the volume by   1      /    a  b  c. So the volume of the ellipsoid is abc times the volume of the unit sphere.Of course that method doesn&#039;t give you any practice with triple integrals.

allucinemsj · Accepted Answer

Step 1You can also use a modified spherical coordinate to describe the surface (not the interior) as                    x                            =        a        cos        ⁡        (        v        )        sin        ⁡        (        u        )                            y                            =        b        sin        ⁡        (        v        )        sin        ⁡        (        u        )                            z                            =        c        cos        ⁡        (        u        )              0  ≤  v  &amp;lt;  2  π and   0  ≤  u  ≤  π.Step 2Then use the vector field   F  =  (  0  ,  0  ,  z  ) and compute the outer flux through the surface. Note that the divergence of F is 1 hence by Divergence(Gauss) theorem it will be equal to the volume. But, surface integral is pretty easy.

Find the volume of the ellipsoid defined by x^2/a^2+y^2/a^2+z^2/a^2 leq 1.

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