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: : 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)