calculating-integrals.md

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Calculating Integrals

Rectangle Rule

• Divide the range [a,b] in N equal width rectangles
• Let m be the midpoint along the width of a rectangle
• Its height is then f(m)
  • This height can be negative, and as such, results in a negative area
• Add the areas of all rectangles
• Larger N results in better approximation

Pseudocode

width = (b-a)/N
area = 0
for i = 0 to N-1:
	mid = a + (i+0.5)*width
	height = f(mid)
	area += height*width
return area

Trapezoidal Rule

• Use trapezoids instead of rectangles
• Are of trapezoid with width w and side lengths u and v:
  • = wv + 0.5w(u-v)
  • = wv + 0.5wu - 0.5wv
  • = 0.5wv + 0.5wu
  • = 0.5w(u+v)

Pseudocode

width = (b-a)/N
area = 0
for i = 0 to N-1:
	start = a + i*width
	u = f(start)
	v = f(start + width)
	area += 0.5*w*(u+v)
return area