3.3 CALCULUS: THE LANGUAGE OF CHANGE

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

• 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.

• 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).

Sir Isaac Newton (1642-1727)

Gottfiried Leibniz (1646-1716)

• 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.

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

1n b,a

121 nx,...,x,x

.bxx...xxxa nn 1210

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

|

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

.I ii

xn-1

|

1 i n

Denote the length of the ith sub-interval by

so that

xiΔ

011Δ xxx

122Δ xxx

233Δ xxx

1Δ iii xxx

1Δ nnn xxx

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

On the ith sub-interval, construct a rectangle of length and whose width is . xiΔ if

x

yy = f(x)

ba| |

x1

| xi-1

|xi

|xn-1

|

1 i n

1f if

nf

The area of the ith rectangular region is

.xiΔ if

Thus, the sum of areas of all n rectangular regions is

xf 11 Δ xf 22 Δ ...xf... ii Δ xf nn Δ

or .xf ii

n

i

Δ1

The area of the region R is given by

.xflim ii

n

inΔ

1

x

y

y = f(x)

ba| |

R

.dxxfxflimb

aii

n

in

1

Δ

Define

Thus, the area of the region R is given by

.dxxfRAb

a

Length of a rectangular element

width of a rectangular element