Calculus is the mathematical study of change. It has two main areas: differential calculus concerns slopes and rates of change, while integral calculus concerns area and volume. The foundations of calculus were established in the late 17th century by Newton and Leibniz, who recognized that differentiation and integration are inverse processes. Calculus is based on the concept of a limit, which allows approximating values that cannot be calculated directly. The document then provides an example of using limits to calculate the area under a curve by dividing it into rectangular elements and taking the sum as the number of elements approaches infinity.

1. 3.3 CALCULUS: THE LANGUAGE OF
CHANGE

2. What is Calculus?
• It is the mathematical study of the dynamics of
change.
• Calculus is the mathematics of slopes and
areas; it is a set of mathematical methods
used to approximate values of slopes and
areas under curves
Two main areas of study:
• Differential Calculus - concerned with slopes
and rates of change of functions
• Integral Calculus - concerned with area and
volume

3. • The beginnings of integration can be
recognized in the work of the ancient
Greeks (Euclid, Archimedes ) in finding
areas of curved regions and volumes of
curved solids.
• The beginnings of differentiation were
much later, in the work of the early 17th
century on tangents to curves and
instantaneous rates of change.

4. • The recognition that differentiation and
integration are inverses of each other
(the "Fundamental Theorem of Calculus")
and the major initial development of the
theory occurred in the late 17th century,
mainly in the work of Newton (1642-1727)
and Leibniz (1646-1716).

5. Sir Isaac Newton (1642-1727)

6. Gottfiried Leibniz (1646-1716)

7. • All calculus was based on the concept of a limit,
which is the destination-value for a sequence
of values.
• With this concept, one can approximate values
that cannot be calculated directly, and use
those approximations to find the precise value.
• This concept was not well understood (because
it uses the concept of infinity) until the 19th
century (in the work of Cauchy, Riemann,
Weierstrass and others)
• Until then the results in the calculus were
founded on an unsound, non- rigorous basis.

8. Area of a plane region
Let f be a function of x which is
continuous and non-negative on the closed
interval
[ ].b,a
Subdivide the closed interval into n sub-
intervals by choosing intermediate numbers
, where
( )1−n
[ ]b,a
121 −nx,...,x,x
.bxx...xxxa nn =<<<<<= −1210

9. For example, if we want to subdivide
[a,b] into 2 sub-intervals, we only choose
one number x1 between a and b.
x1
|
a b
| |
If we want to subdivide [a,b] into 3 sub-intervals,
we choose two numbers x1 and x2 between a and b.
a b
| |
x1
|
x2
|

10. x
y
y = f(x)
ba
| |
x1
|
xi-1
|
xi
|
Denote the ith sub-
interval by Ii so that
[ ]101 x,xI =
[ ]212 x,xI =
[ ]323 x,xI =
[ ]iii x,xI 1−=
[ ]nnn x,xI 1−=
For each ,
choose a number
n,...,,i 21=
.Iii ∈ε
xn-1
|
1ε iε nε

11. Denote the length of the ith sub-interval by
so that
xiΔ
011Δ xxx −=
122Δ xxx −=
233Δ xxx −=
1Δ −−= iii xxx
1Δ −−= nnn xxx

12. Our objective is to
find a formula for
the area of the
region R enclosed
by the graph of f,
the x-axis and the
lines given by x = a
and x = b.
x
y
y = f(x)
ba
| |
R

13. On the ith sub-interval, construct a rectangle of
length and whose width is .xiΔ( )if ε
x
y
y = f(x)
ba
| |
x1
|
xi-1
|
xi
|
xn-1
|
1ε iε nε
( )1εf
( )if ε
( )nf ε

14. The area of the ith rectangular region is
.xiΔ⋅( )if ε
Thus, the sum of areas of all n rectangular regions
is
( ) xf 11 Δε ( ) xf 22 Δε+ ( ) ...xf... ii +++ Δε ( ) xf nn Δε+
or
( ) .xf ii
n
i
Δ
1
ε∑=
The area of the region R is given by
( ) .xflim ii
n
i
n
Δ
1
ε∑=
∞→

15. x
y
y = f(x)
ba
| |
R

16. ( ) ( ) .dxxfxflim
b
a
ii
n
i
n ∫∑ =
=
∞→
1
Δε
Define
Thus, the area of the region R is given by
( ) ( ) .dxxfRA
b
a∫=
Length of a
rectangular
element
width of a rectangular
element