The curve y = 1 / x in the
is rotated about the x-axis to generate a solid.
Torricelli's Trumpet or Gabriel's Horn
Evangelista Torricelli ( 1608 - 1647 ) was a student of Galileo. As a young man he studied in Galileo's home at Arcetri near Florence. Upon Galileo's death, Torricelli succeeded his teacher as mathematician and philosopher for their good friend and patron, the Grand Duke of Tuscany.
Torricelli's own words fully describe his amazement at discovering an infinitely long solid with a surface that calculates to have an infinite area, but a finite volume. "It may seem incredible that although this solid has an infinite length, nevertheless none of the cylindrical surfaces we considered has an infinite length but all of them are finite." This "incredible" paradox prompted Torricelli to try several alternate proofs.
Torricelli was born a bit too
The study of infinitesimal was too new. Recall that Newton was
the year Galileo had died (1642) and Leibniz was yet four years
Unfortunately, Torricelli did not live to see the methods of calculus
emerge to confirm his painstaking calculations largely based on his
Cavalieri's "summation of plane slices" method.