# double integral

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## Presentation Transcript

### Double Integrals :

Double Integrals Introduction

### Volume and Double Integral :

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 = ?

### Slide 4:

Volume of ij’s column: Total volume of all columns: ij’s column: Area of Rij is Δ A = Δ x Δ y

Definition

### Definition: :

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

### Example 1 :

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

### Slide 9:

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

Example 2

### Integrals over arbitrary regions :

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

### Properties :

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

### Slide 13:

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

### More general case :

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 a b x y h(x) g(x) x A

### Similarly :

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 d x y c h(y) g(y) y A

### Note :

Note If f (x, y) = φ (x) ψ(y) then

### Examples :

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