Homework 1 – due January 20th
In the class we defined the definite integral of a continuous function as
Z
lim Ln = lim Rn =
n→∞
n→∞
b
f (x) dx,
a
where Rn and Ln are the right Riemann sum and the left Riemann sum respectively.
In this assignment we’ll consider the definite integral of more general (not necessarily continuous) functions.
Let f (x) be a function (not necessarily continuous) defined on [a, b]. Define the definite integral of f (x)
from a to b as
Z b
n
X
f (x) dx = lim
f (x∗i )∆x,
n→∞
a
b−a
n
i=1
x∗i
is any point in interval [xi−1 , xi ], i=1, 2, . . . n.
(n equal slices) and
with ∆x =
Here the limit does not depend on the points x∗i , i=1, 2, . . . n (very important!) and we call these
functions integrable. When f (x) is a continuous function the above limit doesn’t depend on x∗i , i=1, 2,
. . . n (Theorem 3 in the textbook), i.e. f (x) is integrable. That’s why we usually use only the left and right
Riemann sums when we work with continuous functions.
1. (a) Prove that for
(
1, if x is rational in [0, 1]
f (x) =
,
2, if x is irrational in [0, 1]
limn→∞ Ln = limn→∞ Rn , where Rn and Ln are the right Riemann sum and the left Riemann
sum respectively.
(b) Is f (x) an integrable function on the interval [0, 1]?
2. Express the limit
lim
n→∞
n
X
i=1
n
(i + n)(i + 2n)
as a definite integral of a function on an interval.
1