1. Double Integrals over Rectangles

2. Double Integrals over Rectangles
Consider a closed rectangular region
and 𝑓(𝑥,𝑦) defined on R.
Assume 𝑓(𝑥,𝑦)≥0
Let S be the solid bounded below by the region R
and above by the graph of f.
Goal: Find the volume of S.
Divide the region R into sub-rectangles
This creates rectangles 𝑅 𝑖𝑗 of area ∆𝐴=∆𝑥∆𝑦
On each rectangle pick a sample point

3. Double Integrals over Rectangles
Estimate the volume above 𝑅 𝑖𝑗 by
= volume of post
Add this up for all the rectangles to obtain a double Riemann Sum:
If we take more sub-divisions having a smaller mesh, we expect this sum to get closer to the exact value of the volume.
Theorem: exists if 𝑓(𝑥,𝑦) is continuous

4. Double Integrals over Rectangles – Example 1
Find an approximate value for over 0≤𝑥≤2, 1≤𝑦≤ with m = 2 and n = 2 using the midpoint rule.
Rectangle
Midpoint
f(Midpoint)
R11
(0.5, 1.25)
R12
(0.5, 1.75)
R21
(1.5, 1.25)
R22
(1.5, 1.75)
Note that the result is negative since the surface is below the xy plane on the given region R.

5. Double Integrals over Rectangles - Applications
For a function that takes on both positive and negative values, is a difference of volumes: 𝑉 1 − 𝑉 2 , where 𝑉 1 is the volume above R and below the graph of 𝑓 and 𝑉 2 is the volume below R and above the graph of 𝑓
If f(x, y) is the population density in a region R, then
gives the total population in the region R.
If f(x, y) is the mass density (kg/m2) of a lamina shaped like R, then
gives the total mass (kg) of the lamina.

6. Double Integrals over Rectangles - Iterated Integrals
Suppose 𝑓(𝑥,𝑦) is continuous on the rectangle R: 𝑎≤𝑥≤𝑏, 𝑐≤𝑦≤𝑑
We want to determine algebraically.
First integrate with respect to y (keeping x constant) from y = c to y = d, and then integrate the resulting function of x with respect to x.
We can change the order of integration:
First integrate with respect to x (keeping y constant) from x = a to x = b and then integrate the resulting function of y with respect to y.
Fubini’s Theorem: If 𝑓 is continuous on the rectangle R, then

7. Double Integrals over Rectangles – Example 2
Evaluate where
Method 1: (first integrate w.r.t y, then x)
Method 2: (first integrate w.r.t. x, then y)

8. Double Integrals over Rectangles – Example 3
Find the volume of the solid bounded by the surface and the planes x = 0, x =1, y = 0, y = 1 and z =0.
Method 1: (first integrate w.r.t. y)
𝑧=𝑓(𝑥,𝑦)
𝑦=0 R
𝑥=1
𝑦=1