Finding the volume of solidUse triple integral to find the volume of the solid enclosed between the elliptic cylinder x 2 + 9 y 2 = 9 and the planes z = 0 and z = x + 3

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Finding the volume of solidUse triple integral to find the volume of the solid enclosed between the elliptic cylinder       x    2    +  9      y    2    =  9 and the planes   z  =  0 and   z  =  x  +  3

Dayana Doyle · Accepted Answer

Step 1As with any of these problems, sketch the region. You should get an elliptic cylinder with the z axis as the axis of symmetry, cut in half diagonally by the given plane. In fact, you should at once be able to see that the sought-after volume is simply half of the volume of the elliptic cylinder between   z  =  0 and   z  =  6 -this will serve as a check on your integration result.As for the integral, which you demand be a &quot;triple&quot; integral (if you get more experienced at this, you&#039;ll find that people who actually use these things do not care what you call them), we consider trapezoidal cross-sections parallel to the y axis. Why? Because the height of the region is independent of y. At position x, the width is dx and the height &quot;varies&quot; between   z  =  0 and   z  =  x  +  3.Step 2(I must mention here that this demand for a &quot;triple&quot; integral hobbles the understanding of the problem, in which all we are doing is adding up trapezoidal volume elements, which as you&#039;ll see is just a single integral, which is much easier to understand in the first place.)I hope you see how to derive the limits in x and y: simply consider the boundaries of the elliptic cylinder, which do not change with z.Step 3The triple integral is thus  V  =      ∫          −      3        3    d  x        ∫          −      (      1              /            3      )              9        −                  x          2                            (      1              /            3      )              9        −                  x          2                      d  y        ∫    0          x      +      3        d  zThe inner two integrals trivially evaluate, and you are left with, as I mentioned above, a single integral:  V  =      2    3        ∫          −      3        3    d  x    (  x  +  3  )      9    −          x      2      I hope you can evaluate this. BTW the answer should come out to   9  π

Use triple integral to find the volume of the solid enclosed between the elliptic cylinder x^2+9y^2=9 and the planes z=0 and z=x+3.

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