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,