example 1

"Riemann_1.gif"

"Riemann_2.gif"

Left endpoint

"Riemann_3.gif"

"Riemann_4.gif"

Right endpoint

"Riemann_5.gif"

"Riemann_6.gif"

Midpoint

"Riemann_7.gif"

"Riemann_8.gif"

example 2

"Riemann_9.gif"

"Riemann_10.gif"

Left endpoint

"Riemann_11.gif"

"Riemann_12.gif"

Right endpoint

"Riemann_13.gif"

"Riemann_14.gif"

Midpoint

"Riemann_15.gif"

"Riemann_16.gif"

example 3

"Riemann_17.gif"

"Riemann_18.gif"

Left endpoint

"Riemann_19.gif"

"Riemann_20.gif"

Right endpoint

"Riemann_21.gif"

"Riemann_22.gif"

Midpoint

"Riemann_23.gif"

"Riemann_24.gif"

example 4

"Riemann_25.gif"

"Riemann_26.gif"

Left endpoint

"Riemann_27.gif"

"Riemann_28.gif"

Right endpoint

"Riemann_29.gif"

"Riemann_30.gif"

Midpoint

"Riemann_31.gif"

"Riemann_32.gif"

Spikey Created with Wolfram Mathematica 7.0