14.6 Triple Integrals 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. 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 complex four-space. Taken from: http://emsh.calarts.edu/~mathart/sw/Color_3D_Prints.html

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

Example 5 Solution 2

Example 1 Evaluate the triple iterated integral

Solution Example 1

Example 2 Find the volume of the ellipsoid given by

Solution Example 2

Example 3 Evaluate the given integral (Hint: change the order of integration)

Example 3 solution

Figure 14.52

Figure 14.59