1. Rules of Integration
By
Oladokun Sulaiman Olanrewaju

2. Area Under A Curve
• The area under a curve is the area bounded by
the curve y = f(x), the x-axis and the vertical
lines x = a and x = b.
y y = f(x)
Suppose we
want to know
the area of the
shaded region
x
x=a x=b

3. Area and the Definite Integral
• If a function has only positive values in an
interval [a,b] then the area between the curve
y=f(x) and the x-axis over the interval [a,b] is
expressed by the definite integral,
b
a
f ( x)dx
• It is called the definite integral because the
solution is an explicit numerical value.

4. f(x) < 0 for some interval in [a,b]
b

a
f ( x)dx  Area of R1  Area of R2  Area of R3
y
y = f(x)
R3
R1
x
R2
x=a x=b

5. y
To find
the area y = f(x)
below
the x-
axis R3
R1 place a
negative
sign x
R2
before
the x=b
x=a x=c
integral x=d
Alternatively,
Area of R1  Area of R2  Area of R3
c d b
  f ( x)dx   f ( x)dx   f ( x)dx
a c d

6. Area Between Two Curves
Let f and g be continuous functions such that f(x)>g(x)
on the interval [a,b]. Then the area of the region
bounded above by y=f(x) and below by y=g(x) on [a,b]
is given by y
y = g(x)
 f ( x)  g ( x)dx
b
a y = f(x)
x
x=a x=b

7. Fundamental Theorem of Calculus
• If f is continuous on [a,b], then the definite
integral is
b
a
f ( x)dx  F (b)  F (a)
where F(x) is any antiderivative of f on [a,b]
such that F ' ( x)  f ( x).

8. Evaluating Definite Integrals
b
To find  f ( x)dx
a
First find the indefinite Integral  f ( x)dx F ( x)  C
Then find F (a) and F (b).
b
b
Finally,  f ( x)dx  F ( x)  F (b)  F (a)
a
a

9. Consumer Surplus
Price, p Consumer surplus is a measure
of consumer welfare. The area
of this shaded region is
x*
Consumer 0
D( x)dx  p * x *
Surplus
p = p*
Demand function
p = D(x)
0 x* Quantity, x

10. Producer Surplus
Price, p Supply function
p = S(x) Producer surplus is the area of
this shaded region which is
x*
p * x *  S ( x)dx
0
p = p*
Producer
Surplus
Demand function
p = D(x)
0 x* Quantity, x