Evaluate the following integrals as a limit of sums:

Formula used:

where,

Here, a = 0 and b = 2

Therefore,

Let,

Here, f(x) = x² + 2 and a = 0

Now, by putting x = 0 in f(x) we get,

f(0) = (0)² + 2 = 0 + 2 = 2

f(h)

= (h)² + 2

Similarly, f(2h)

= (2h)² + 2

Since 2 is repeating n times in series

Now take h² common in remaining series

Put,

Since,