Multiple Integrals

 Objective Questions

Double and Triple Integrals

Partition of a rectangle

In the xy-plane, consider a rectangle bounded by the lines x = a, x = b; y = c, y = d, a < b, c < d.

The region within this rectangle including its boundary be denoted by [a, b; c, d] or R. Let us consider the area to be the number (b - a) (d - c).

Let P₁ = {a = x₀, x₁, x₂, …,x_n= b} and P₂ = {c = y₀, y₁, y₂, …,y_m= d}be the partitions of the intervals [a, b] and [c, d] respectively. Lines drawn parallel to the axes through the points of partitions P₁ and P₂ give rise to a partition P of the rectangle R into mn sub-rectangles

                [x_i-1, x_i; y_j-1, y_j], i = 1, 2, …, n and j = 1, 2, …,m.

The partition P in fact Cartesian product of P₁ and P₂, where

                P = P₁ ´ P₂ = {(x_i, y_j); x_iÎP₁ and y_jÎP₂}

We shall use the symbol DR_ij to denote the  sub-rectangle [x_i-1, x_i; y_j-1, y_j] and its area by (x_i - x_i-1) (y_jm- y_j-1) = Dx_i Dy_j

m(P₁) = Dx_r  and m(P₂) = Dy_s are norms of P₁ and P₂.

DR_rs = [x_r-1, x_r; y_s-1, y_s] is the norm of the partition of rectangle.

Each DR_ij ® 0 as m(P)®0.

Refinement (finer)

The partition P₁ is said to be refinement of P₂ if P₁ Ê P₂.

Integration over a rectangle

Let f be a bounded function of x, y over a rectangle R = [a, b; c, d]. Let P be partition of R which divides R into mn sub-rectangles DR_ij, i = 1, 2, …, n and j = 1, 2, …,m.

Let M _ij and m_ij be the upper and lower  bounds of f in DR_ij.

Consider two sums

a