1. Chapter 15 – Multiple Integrals 15.1 Double Integrals over Rectangles 1 Objectives:  Use double integrals to find volumes  Use double integrals to find average values

2. Definite Integral Review First, let’s recall the basic facts concerning definite integrals of functions of a single variable: If we divide an interval into subintervals of the same width, then we form the Riemann Sum If we take the limit of this then we obtain the definite integral 15.1 Double Integrals over Rectangles2

3. Volumes In a similar manner, we consider a function f of two variables defined on a closed rectangle R = [a, b] x [c, d] = {(x, y)  R 2 | a ≤ x ≤ b, c ≤ y ≤ d} and we first suppose that f(x, y) ≥ 0. The graph of f is a surface with equation z = f(x, y). 15.1 Double Integrals over Rectangles3

4. Volumes Let S be the solid that lies above R and under the graph of f, that is, S = {(x, y, z)  R 3 | 0 ≤ z ≤ f(x, y), (x, y)  R} Our goal is to find the volume of S. 15.1 Double Integrals over Rectangles4

5. Volumes We can approximate the part of S that lies above each R ij by a thin rectangular box (or “column”) with: ◦ Base R ij ◦ Height f (x ij *, y ij *) The volume of this box is the height of the box times the area of the base rectangle: f(x ij *, y ij *) ∆A 15.1 Double Integrals over Rectangles5

6. Volumes We follow this procedure for all the rectangles and add the volumes of the corresponding boxes. Thus, we get an approximation to the total volume of S: 15.1 Double Integrals over Rectangles6

7. Volumes Our intuition tells us that the approximation becomes better as m and n become larger. So, we would expect that: 15.1 Double Integrals over Rectangles7

8. Volumes This double sum means that: ◦ For each subrectangle, we evaluate f at the chosen point and multiply by the area of the subrectangle. ◦ Then, we add the results. 15.1 Double Integrals over Rectangles8

9. Definition – Double Integral 15.1 Double Integrals over Rectangles9

10. Double Integral The sample point (x ij *, y ij *) can be chosen to be any point in the subrectangle R ij *. However, suppose we choose it to be the upper right-hand corner of R ij [namely (x i, y j )]. Then, the expression for the double integral looks simpler: 15.1 Double Integrals over Rectangles10

11. Volume 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: 15.1 Double Integrals over Rectangles11

12. Example 1 If R= [-1,3] x [0,2], use a Riemann sum with m = 4, n=2 to estimate the value of. Take the sample points to be the upper left corners of the squares. 15.1 Double Integrals over Rectangles12

13. Mid Point Rule 15.1 Double Integrals over Rectangles13

14. Average Value We define the average value of a function f of two variables defined on a rectangle R to be 15.1 Double Integrals over Rectangles14

15. Example 2 – pg.982 #12 Evaluate the double integral by first identifying it as the volume of a solid. 15.1 Double Integrals over Rectangles15

16. More Examples The video examples below are from section 14.6 in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length. ◦ Example 1 Example 1 ◦ Example 2 Example 2 ◦ Example 3 Example 3 15.1 Double Integrals over Rectangles16