1. Riemann Sums
Consider any function f (x) where a ≤ x ≤ b.
b−a
Partition [a, b] into n subintervals of equal length ∆x =
n
Evaluate the function f (x) at sample points c1 , c2 , ... cn
chosen inside each subinterval.
Y
Positive values
a x0 x1 x2 x3 x4 x5 b x6
c1 c2 c3 c4 c5 c6
X
Negative values
Form the Riemann sum:
f (c1 ) · ∆x + f (c2 ) · ∆x + ... + f (cn ) · ∆x

2. Riemann Sum: [f (c1 ) + f (c2 ) + ... + f (cn )] · ∆x
Y
Positive values
a x0 x1 x2 x3 x4 x5 b x6
c1 c3 c4 c5 c6
O c2 X
Negative values

3. The Definite Integral
Notation:
n
f (c1 ) · ∆x + f (c2 ) · ∆x + ... + f (cn ) · ∆x = f (ci ) · ∆x
i=1
Consider more and more subintervals. In other words, let the
number n be larger and larger:
Y
X
Then the limit of the Riemann sums, as the number of
intervals becomes larger and larger is called The Definite
Integral or The Riemann Integral of f (x) from a to b:
b n
f (x) dx = lim f (ci ) ∆x
a n→∞
i=1

4. Geometric Interpretation of the Definite Integral
If f (x) ≥ 0 , which means that the graph of f (x) is above
the x - axis:
b
f (x) dx = area under the graph and above the x axis
a
y
x

5. Geometric Interpretation of the Definite Integral
If f (x) is not positive (or negative) which means that the
graph of f (x) has parts both above and below the x axis:
b
f (x) dx = area above - area below the x -axis
a
y
A1
x
A2

6. Geometric Interpretation of the Definite Integral
More generally:
y
x
b
f (x) dx = area above - area below the x -axis
a
b
f (x) dx = yellow area - blue area
a