




_{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 x_{0} = a to x_{n} = 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 x_{0} = a to x_{n} = 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 x_{0} = a to x_{n} = 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 x_{0} = a to x_{n} = b