Q. 115.0( 1 Vote )

Evaluate the following integrals as a limit of sums:


Answer :


Formula used:



where,



Here, a = 1 and b = 2


Therefore,




Let,



Here, f(x) = x2 – 1 and a = 1



Now, by putting x = 1 in f(x) we get,


f(1) = 12 – 1 = 1 – 1 = 0


f(1 + h)


= (1 + h)2 – 1


= h2 + 12 + 2(h)(1) – 1


= h2 + 2(h)


Similarly, f(1 + 2h)


= (1 + 2h)2 – 1


= (2h)2 + 12 + 2(2h)(1) – 1


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


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




Now take h2 and 2h common in remaining series





Put,



Since,
















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