Evaluate the following integrals as a limit of sums:

Formula used:

where,

Here, a = 0 and b = 2

Therefore,

Let,

Here, f(x) = x² + x and a = 0

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

f(0) = 0² + 0 = 0 + 0 = 0

f(h)

= (h)² + (h)

= h² + h

Similarly, f(2h)

= (2h)² + (2h)

Now take h² and h common in remaining series

Put,

Since,