Evaluate:

5
∫ 3^{y} dy
0

where [p] is the greatest integer ≤ p, and {p} = p - [p]

3^{y} is a sawtooth function, repeating every 1 unit change in y.

So Integral, 0 to 5, of 3^{y} dy is 5 * Integral, 0 to 1, of 3^y dy, as y={y} in that interval.

3^y = e^(y*ln(3))

so

Integral, 0 to 1, of 3^y dy = Integral, 0 to 1, of e^(y*ln(3)) dy = [(3^y) / ln3] evaluated from 0 to 1 = 3/ln3 - 1/ln3 = 2/ln3.

Five times that is 10/ln3 ~= 9.102392266268373

 

Posted by Charlie on 2007-10-12 11:06:04