integral - Integration with Riemann Sum Python -

i have been trying solve integration riemann sum. function has 3 arguments a,b,d lower limit b higher limit , d part a +(n-1)*d < b. code far but. output 28.652667999999572 should 28.666650000000388. if input b lower has calculate have solved problem already.

def integral(a, b, d):     if > b:         a,b = b,a     delta_x = float((b-a)/1000)     j = abs((b-a)/delta_x)     = int(j)     n = s = 0     x =     while n < i:         delta_a = (x**2+3*x+4) * delta_x         x += delta_x         s += delta_a          n += 1      return abs(s)  print(integral(1,3,0.01)) 

there no fault here, neither algorithm nor code (or python). riemann sum approximation of integral , per se not "exact". approximate area of (small) stripe of width dx, between x , x+dx, , f(x) area of rectangle of same width , height of f(x) it's left upper corner. if function changes it's value when go x x+dx area of rectangle deviates true integral.
have noticed, can make approximation closer making thinner , thinner slices, @ cost of more computational effort , time. in example, function f(x) = x^2 + 3*x + 4, , it's exact integral on x in [1.0,3.0) 28 2/3 or 28.66666...

the approximation rectangles crude one, cannot change that. change time takes code evaluate, say, 10^8 steps instead of 10^3. @ code:

def riemann(a, b, dx):     if > b:         a,b = b,a     # dx = (b-a)/n     n = int((b - a) / dx)     s = 0.0     x =     in xrange(n):         f_i = (x + 3.0) * x + 4.0         s += f_i         x += dx     return s * dx   

here, i've used 3 tricks speedup, , 1 greater precision. first, if write loop , know number of repetions in advance use for-loop instead of while-loop. it's faster. (btw, loop variables conventionally i, j, k ... whereas limit or final value n). secondly, using xrange instead of range faster users of python 2.x. thirdly, factorize polynoms when calculating them often. should see code mean here. way, result numerically stable. last trick: operations within loop not depend on loop variable can extracted , applied after loop has ended. here, final multiplication dx.