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Many applications such as Area, Vol, ee ee, popes etc. are under the theory of multiple, ‘@ouble and triple) integration. {, _ These applications are very useful in Engineering field., , . necessary, the study of multiple integral in, , , , TECGme earn), , _s2afnion, , b d(x), An expression of the form J J f(x, y) dxdy is called, a o(x), double integral and it is evaluated in order . That means, for order dxdy integral is evaluated with respect to x, first by keeping variable y constant and then it is, evaluated with respect to y in this order y is called, outside variable of integral and its limit a and b is, always constant., Variable x is called inside variable and its limits ¢(x), , and d(x) is either constant or depends on inside variable, x, , 8.2.1 Solved Examples on Double Integrals, Ex. 8.2.1, , a, Evaluate j, 0, , , , , , , , fo J Myavex= ry :, Ly wi el, Ex. 8.22 ° : :, , DL Geos, pons Va-a0-¥)

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Tt Topic : Change of the Order of, , Integration, , , , , , , valuation of Double Integral by, Changing the Order of Integration, , Sometimes a double integral [Jt y) dx dy is, A, difficult or even impossible to integrate with given |, order of integration. In such cases, it becomes, necessary to change the order of integration in double, integral. :, - Tochange the order of integration first draw the rough, _ .. sketch of the region of integration, find out which strip, is given and the then change the strip, (i.e. if given strip is horizontal then take vertical strip, and vice versa) and find out the new limits an evaluate, the integrals., , Solved Examples : aS