1. Double Integrals over Rectangles
Section 16.1
Double Integrals over Rectangles

2. A CLOSED RECTANGLE
A closed rectangle in the plane is the region given by

3. VOLUME OVER A RECTANGLE
Consider the function f defined on a closed rectangle R and that f (x, y) ≥ 0. The graph of f is a surface with equation z = f (x, y). Let S be the solid that lies above R and under the graph of f, that is
Our goal is to find the volume of the solid S.

4. PARTITIONING THE RECTANGLE
Partition [a, b] into m equal subintervals [xi − 1, xi] of equal width Δx = (b − a)/m.
Partition [c, d] into n equal subintervals [yj − 1, yj] of equal width Δy = (c − d)/n.
Create subrectangles Rij, each of area ΔA = Δx Δy as follows

5. VOLUME OVER A RECTANGLE
To find the volume over a close rectangle R:
Choose a sample point in each Rij.
Find the volume in the rectangular column formed by Rij, that is
Sum all the volumes and take the limit.

6. THE DOUBLE INTEGRAL The double integral of f over the rectangle R is
if the limit exits.
NOTE: The function f does not have to be positive. Its graph can be below the xy-plane.

7. For each subrectangle Rij, if we choose the sample point to be (xi, yj), then the expression for the double integral simplifies to

8. VOLUME AND DOUBLE INTEGRALS
If f (x, y) ≥ 0, then the volume V of the solid that lies above the rectangle R and below the surface z = f (x, y) is

9. DOUBLE RIEMANN SUM The sum
is called the double Riemann sum and is used to approximate the value of the double integral.

10. MIDPOINT RULE FOR DOUBLE INTEGRALS
where is the midpoint of [xi − 1, xi] and
is the midpoint of [yj − 1, yj].

11. AVERAGE VALUE
The average value of a function f of two variables defined on a rectangle R is
where A(R) is the area of R.

12. PROPERTIES OF THE DOUBLE INTEGRAL
If f (x, y) ≥ g(x, y) for all (x, y) in R, then