Riemann Integration

Question

Show that the function f(x):

f(x) = 0 if 0<x<=1

f(x) = 1 if x=0

is Riemann integrable, and integration of f(x) over 0-1, and equals to 0

Answer 1

from 0<x<1 f(x)=0

for x=0, f(x) = 1

break it up into n=4 rectangles. Each has base = 1/4

each has height = f(1/4)=0, f(1/2)=0 f(3/4) = 0 and either f(0)=1 or f(1)=0

total area =sum of rectangles = either 0 for the right side heights, or if you use the left side heights area = 1/4

use the midpoints and it's also 0=area

check the answer(s). Find the definite integral from x=0 to 1

integral of f(x) = F(x) = c. F(1)-F(0) = c - c = 0

or graph the step function. Area under the line =0

area under the point at x=0 is height times base: 1(0)=0

or add in the "area" if F(0) -F(0) = 1-1 =0 for the point where x=0.

the left side Reimann integral overstates the area

the left side and midpoint Reiman approximations are exactly correct. But as n approaches infinity, the left side Reimann integral also approaches zero as the area.