I remember calculating the area of rectangles under the curve to approximate the area under the curve.  You need to make the rectangles narrower and narrower to get closer to the true answer.  You can take f(x) anywhere in the interval (midpoint is usually the most accurate, but let's do the left edge for convenience).

To approximate the integral of f(x) from a to b,
sum(delta-x * f(x)) for a<=x<b, increase x by delta-x each time.

To approximate the integral of x² from 0 to 5,
sum(delta-x * x²) for 0<=x<5, step by delta-x.

For delta-x = 1, sum = 1*0² + 1*1² + 1*2² + 1*3² + 1*4² 
= 0 + 1 + 4 + 9 + 16 = 30

For delta-x = 0.5, sum = 0.5*0² + 0.5*0.5² + 0.5*1² + 0.5*1.5² + 0.5*2² + 0.5*2.5² + 0.5*3² + 0.5*3.5² + 0.5*4² + 0.5*4.5² 
= 0 + 0.125 + 0.5 + 1.125 + 2 + 3.125 + 4.5 + 6.125 + 8 + 10.125 
= 35.625

For delta-x = 0.1, the sum is 40.425
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Hi, thanks for replying. I did a bit of digging and managed to come up with a solution using Rieman sums and the limit definition for the integral of f(x)dx where f(x) = x^2, and I calculated the integral between 0 and 5. Here is the calculation (two screenshots)