double integral

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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

Slide 5: 


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 : 

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)