Double Integrals

1. Double Integrals Introduction

2. Volume and Double Integral z=f(x,y) ≥ 0 on rectangle R=[a,b]×[c,d] S={(x,y,z) in R3 | 0 ≤ z ≤ f(x,y), (x,y) in R} Volume of S = ?

4. ij’s column: z y (xi, yj) Rij f (xij*, yij*) y Sample point (xij*, yij*) x x Δ x Δ y Area of Rijis ΔA = Δ x Δ y Volume of ij’s column: Total volume of all columns:

5. Definition

6. Definition: The double integral of f over the rectangle R is if the limit exists Double Riemann sum:

7. Note 1. If f is continuous then the limit exists and the integral is defined Note 2. The definition of double integral does not depend on the choice of sample points If the sample points are upper right-hand corners then

8. Example 1 z=16-x2-2y2 0≤x≤2 0≤y≤2 Estimate the volume of the solid above the square and below the graph

9. m=n=16 m=n=4 m=n=8 V≈46.46875 V≈41.5 V≈44.875 V=48 Exact volume?

10. z Example 2

11. Integrals over arbitrary regions A • A is a bounded plane region • f (x,y) is defined on A • Find a rectangle R containing A • Define new function on R: f (x,y) 0 R

12. Properties Linearity Comparison If f(x,y)≥g(x,y) for all (x,y) in R, then

13. Additivity A2 A1 If A1 and A2 are non-overlapping regions then Area

14. y d c x a b Computation • If f (x,y) is continuous on rectangle R=[a,b]×[c,d] then double integral is equal to iterated integral y fixed fixed x

15. More general case • If f (x,y) is continuous onA={(x,y) | x in [a,b] and h (x) ≤ y ≤ g (x)} then double integral is equal to iterated integral y g(x) A h(x) x a x b

16. Similarly • If f (x,y) is continuous onA={(x,y) | y in [c,d] and h (y) ≤ x ≤ g (y)} then double integral is equal to iterated integral y d A y g(y) h(y) c x

17. Note If f (x, y) = φ (x) ψ(y) then

18. Examples where A is a triangle with vertices(0,0), (1,0) and (1,1)