Multivariate Calculus - Bruce E. Shapiro

128 LECTURE 16. DOUBLE INTEGRALS OVER RECTANGLES
We then defined Riemann Integral as the limit
∫ b
a
f(x)dx =
lim
δ 1 ,δ 2 ,...,δ n→0,n→∞
n∑
f(c i )δ i
i=1
Figure 16.2: Left: A Riemann Sum in 3D can also be used to estimate the volume
between a surface described some some function f(x, y) and a rectangle beneath it
in the xy-plane. Right: construction of the boxes of height f(u i , v i ) with base in
rectangle i.
We now want to generalize this procedure to find the volume of the solid between
the surface z = f(x, y) and the x-y plane over some rectangle R as illustrated in
figure 16.2. The surface z = f(x, y) forms the top of the volume, and rectangle
R = [a, b] × [c, d]
in the x-y plane forms the bottom of the volume. We can approximate this volume
by filling it up with rectangular boxes, as illustrated on the right-hand sketch in
figure 16.2.
1. Divide the [a, b] × [c, d] rectangle in the x-y plane into little rectangles. Although
they are illustrated as squares in the figure, they do not have to be
squares.
2. Number the little rectangles in the xy plane from i = 1 to i = n.
3. Let the area of rectangle i be ∆A i
4. Pick one point in each rectangle and label it (u i , v i ). The i th point does
not have to be in the center of mini-rectangle i, just somewhere within the
rectangle. The points in rectangle i and rectangle j can be in different locations
within the rectangle.
5. The distance of the shaded surface about the point u i , v i is f(u i , v i ). Draw a
box of height f(u i , v i ) whose base is given by rectangle i.
Revised December 6, 2006. Math 250, Fall 2006