MATH 101: COMPUTING INTEGRALS FROM THE
DEFINITION
LIOR SILBERMAN
In this note I collect a few examples of computing integrals from the definition.
´1
Problem 1. Compute 0 ex dx using the definition. You may use the summation
Pn
n
−q
formula i=1 q i = qq−1
.
Solution: Dividing [0, 1] to n subintervals of length ∆x = n1 , we get the points
xi = ni where f (xi ) = ei/n for f (x) = ex . Thus using the right-side rule we
calculate:
n
n
X
1
1 X 1/n i
ei/n
lim
= lim
e
n→∞
n→∞ n
n
i=1
i=1
n
1 e1/n − e1/n
= lim
(formula with q = e1/n )
n→∞ n
e1/n − 1
e − e1/n
1/n n
= lim
(e
)
=
e
n→∞ n(e1/n − 1)
limn→∞ e − e1/n
=
.
1/n
limn→∞ e 1/n−1
as n → ∞,
1
n
→ 0 so we need to compute
limx→0 (e − ex )
.
x
limx→0 e x−1
In the numerator there is no problem: limx→0 (e − ex ) = e − e0 = e − 1. In the
denominator, we recognize the limit as the definition of the derivative of f (x) = ex
(0)
at x = 0. Since f 0 (x) = ex we have limx→0 f (x)−f
= f 0 (0) = 1 so
x−0
limx→0 (e − ex )
e−1
=
,
x
1
limx→0 e x−1
and
ˆ
1
ex dx = e − 1 .
0
´b
Problem 2. Compute a x dx using the definition. You may use the summation
Pn
formula i=1 i = n(n+1)
.
2
This note is specifically excluded from the terms of UBC Policy 81.
1
MATH 101: COMPUTING INTEGRALS FROM THE DEFINITION
2
Solution: Dividing [a, b] to n subintervals of length ∆x = b−a
points
n , we get the
b−a
xi = a + b−a
i.
We
are
working
with
f
(x)
=
x
so
f
(x
)
=
x
=
a
+
i
.
Thus
i
i
n
n
using the right-side rule we calculate:
!
n
n n
X
X
b−a X
b−a b−a
b−a
i
= lim
a+
i
lim
a+
n→∞
n→∞
n
n
n
n
i=1
i=1
i=1
!
n
n
X
b−a
b−aX
a
i
= lim
1+
n→∞
n
n i=1
i=1
b−a
b − a n(n + 1)
= lim
a·n+
·
n→∞
n
n
2
b−a
b − a b − a n(n + 1)
= lim
· (an) +
·
·
n→∞
n
n
n
2
2
1
1
(b − a)
= lim (b − a)a +
1+
lim
=0
n→∞
n→∞ n
2
n
(b − a)2
b−a
= (b − a)a +
= (b − a) a +
2
2
2a + b − a
b+a
= (b − a)
= (b − a)
2
2
b2 − a2
=
.
2
In other words:
ˆ b
b2 − a2
x dx =
.
2
a
Remark. If 0 < a < b the area under the graph is a trapeze and you can compute
it geometrically. Of course, the integral can also be computed using the FTC and
integration techniques. But if you are asked to use the definition then you can only
get credit for a solution that computes the limit of the Riemann sums as above.
Problem 3 (Challenge). Try computing
i
(Hint: ea+i∆x = ea · e∆x ).
´b
a
ex dx using the techniques above.