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,