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14.6 Triple Integrals

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Andrew Hanson has made some pictures, and I have in turn made sculpture, of a system analogous to Fermat's last theorem - a superquadric surface parameterized complexfour-space.

Taken from: http://emsh.calarts.edu/~mathart/sw/Color_3D_Prints.html

Seventeenth-Century French mathematician Pierre de Fermat wrote in the margin of his copy of Arithmetica by Diophantus, near the section on the Pythagorean Theorem (a squared plus b squared equals c squared), "x ^ n + y ^ n = z ^ n - it cannot be solved with non-zero integers x, y, z for any exponent n greater than 2. I have found a truly marvelous proof, which this margin is too small to contain." This was left as an enigmatic riddle after Fermat's death and it became a famous, unsolved problem of number theory for over 350 years.

Find the area of the region by using the integration order dy dx

Evaluate the triple iterated integral

Find the volume of the ellipsoid given by

Evaluate the given integral (Hint: change the order of integration)