For a functions of one variable, a definite integral represents the area underneath its two-dimensional
function graph. Likewise, the definite integral of a function of two variables represents the volume underneath its
three-dimensional function graph.
Demos
Approximation by Rectangular Prisms
Exercises
For which of the following two functions is the lower sum a better approximation for the domain 0 ≤ x ≤ 1, 0 ≤ y ≤ 1? Explain your answer.
f (x, y) = x2 + y2
f (x, y) = –(x2 + y2)
What would happen if there were a large number of subdivisions (approaches infinity) along the x-axis and a small number of subdivisions along the y-axis? How could you go about finding the sum of the volumes of the rectangular prisms? What if instead there were a large number of subdivisions along the y-axis and a small number of subdivisions along the x-axis?
