Evaluate the following integrals as a limit of sums:
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,