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
