Evaluate the following integrals as a limit of sums:


Formula used:



where,



Here, a = 2 and b = 3


Therefore,




Let,



Here, f(x) = x2 and a = 2



Now, by putting x = 2 in f(x) we get,


f(2) = 22 = 4


f(2 + h)


= (2 + h)2


= h2 + 22 + 2(h)(2)


= h2 + 4 + 4(h)


Similarly, f(2 + 2h)


= (2 + 2h)2


= (2h)2 + 22 + 2(2h)(2)


= (2h)2 + 4 + 4(2h)


{ (x + y)2 = x2 + y2 + 2xy}




In this series, 4 is getting added n times



Now take h2 and 4h common in remaining series





Put,



Since,
















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