Multiple Integrals

An expression of the form ∫_a^b { ∫_f1(x) ^f2(x) f(x,y)dy}dx  is called repeated integral.

To find its value, we first integrated f(x, y) with respect to regarding x as a constant, and then substitute the limits for y. The resulting function is then integrated with respect to x in the usual way. Generally the brackets in the above integral are omitted and the integral is written

            ∫_a^b{ ∫_f1(x)^f2(x) f(x,y)dy}dx                             … (1)

Here the first integration is with respect to y and the second with respect to x.
The region ABCD bounded by y = f₁(x),y = f₂(x),x = a and x = b is called the region of integration.


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