Double Integrals Introduction

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 = ?

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

Definition

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

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

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

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

Example 2 z

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

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

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

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

More general case If f (x,y) is continuous on A={(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

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

Note If f (x, y) = φ (x) ψ(y) then

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