Use a surface integral to find the area of the triangle t in double-struck r3 with vertices at (1, 1, 0), (2, 1, 2), and (2, 3, 3). verify your answer by finding the lengths of the sides and using classical geometry.

Parameterize the triangle by




where . Then the surface element is



and the area of the triangle is



To confirm this result, we can determine the length of each side of the triangle - they are , , and - then apply Heron's formula. If is the semiperimeter, then , and the area is



as expected.

Wow. that is a really cool question.