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Rectangular rule: left endpoints
The curve is approximated on each interval by a constant: f(x k - 1 ).
By adding up the area of all the rectangles from x0 = a to xn = b
Rectangular rule: right endpoints
The curve is approximated on each interval by a constant: f(x k ).
By adding up the area of all the rectangles from x0 = a to xn = b
Trapezoidal rule
The curve is approximated on each interval by a straight line passing through f(x k - 1 ) and f(x k ).
By adding up the area of all the trapezoid from x0 = a to xn = b
Simpson's rule
The curve is approximated on each interval by a portion of parabola passing
through f(x k - 1 ), f(x k ), f(x k + 1 ).
By adding up the area under all the parabolic arcs from x0 = a to xn = b
