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) = x^{2} – 1 and a = 1

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

f(1) = 1^{2} – 1 = 1 – 1 = 0

f(1 + h)

= (1 + h)^{2} – 1

= h^{2} + 1^{2} + 2(h)(1) – 1

= h^{2} + 2(h)

Similarly, f(1 + 2h)

= (1 + 2h)^{2} – 1

= (2h)^{2} + 1^{2} + 2(2h)(1) – 1

= (2h)^{2} + 2(2h)

{∵ (x + y)^{2} = x^{2} + y^{2} + 2xy}

Now take h^{2} and 2h common in remaining series

Put,

Since,

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