Evaluation of Double Integrals

Double integrals over a region may be evaluated by two successive integrations.

Let y1, y2 be functions of x and x1, x2 be constant. f(x, y) is first integrated w.r.t y keeping x fixed between limits y1, y2. Then, the resulting expression is integrated w.r.t.x within the limits x1, x2, i.e.,

            I= ∫_x1^x2 ∫_y1^y2  f(x,Y)dy  dx

Here, integration is carried from the inner to the outer rectangle. Here AB and CD are the two curves whose equations are y1 = f1(x) and y2(x). PQ is a vertical of width dx.

Then the inner rectangle integral means that the integration is along once edge of the strip PQ from P to Q (x remaining constant). The outer rectangle integral corresponds to the sliding of the edge from AC to BD.
Thus the whole region of integration is the area ABCD.



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