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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= ∫x1x2y1y2  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.

Evaluation of Double Integrals

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