Definite Integrals Sums Math 101

7/30/2019 Definite Integrals Sums Math 101

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Outline Readings and exercises Area of a circular disk Areas of rectangles Area of a triangle D

Riemann sums

S Sial

Dept of Mathematics

LUMS

15 November 2011
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Outline

1 Readings and exercises

2 Area of a circular disk3 Areas of rectangles

4

Area of a triangle5 Definite integrals

6 Definite integrals via Riemann sums
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Readings and exercises

Strang 5.1
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Archimedes and the circle

Archimedes

287 BCE - 212 BCE (about)hydrostatics, principle of the lever,

advanced weaponry

proto-integrationdeath claw?

death ray?!!!!??
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Archimedes and the circle contd

Thought we were doing antiderivatives?

Well see the connection a little later.
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O tli R di d i A f i l di k A f t l A f t i l D


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ParabolaAlso found the area under a parabola.

O tli R di d i A f i l di k A f t l A f t i l D
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And then ...the Greek kingdoms fell and the world forgot.

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Area of a rectangle

Rectangle

between x = 0, x = 1, y = 0, y = 1.

Whats the area?

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Area of a rectangle contd

Rectangle between x = 0, x = w, y = 0,

y = h

A = hw

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Area of two rectangles resting onthe x-axis

Rectangles

left side right side height

x = 0 x = w y = wx = w x = 2w y = 2w

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Area of two rectangles contd

Area

A = w w + w 2w = 3w2

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Area of n rectangles

Rectangles

each of width w

bases of rectangles: [0, w], [w, 2w], ... ,

[(n 1)w, nw]heights: w, 2w, ... , (n 1)w, nw

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g g g

Area of n rectangles contd

Area

A = w2 + 2w2 + 3w2 + + nw2

= (1 + 2 + + n)w2

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g g g


Identity

1 + 2 + + n =n(n + 1)

2

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Area

A = (1 + 2 + + n)w2

=

n(n + 1)

2 w2

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Area of a triangle

Triangle bounded byy = 0

y = x

x = 1

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Area of a triangle contd

Rectanglesn rectangles of the same width

base of rectangles covers [0, 1]

top right corner touches line y = x

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Width of each rectangle

w =1 0

n=

1

n

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Bottom corners of rectangles

P =

0,

1

n,

2

n, ,

n

n

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Heights of the rectangles

1

n,

2

n, ,

n

n

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Area of the kth rectangle

Ak = hkwk =k

n

1

n

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Total area of the rectangles

A =

n

k=1

height of rectangle

k

n 1

nwidth of rectangle

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Total area of the rectangles contd

A =

nk=1

kn 1

n

= 1n2

nk=1

k

=

1

n2

n(n + 1)

2 =

1

2 +

1

2n



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In the limit

Atriangle = limn

1

2+

1

2n=

1

2



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Area under a curve

Problem

f : [a, b] R

Find the area between the graph of f and

the x-axis from a to b. Areas below the axisare taken to be negative.

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Area under a curve contd

Notation x=b

x=a

f(x) dx



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Example

x=4

x=1(x3 10x + 1) dx means this area



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Example

The integral we evaluated was

x=1

x=0

x dx =1

2

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Riemann sums

Evaluate x=b

x=a

f(x) dx

if the definite integral exists.

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Riemann sums contd

Partition

P = {x0 = a, x1, x2, xn = b}

xk divide [a, b] into n subintervals.

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Riemann sums contd

Partition contd

||P|| = max {xi+1 xi}



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Riemann sums contd

Width of the ith interval

i = xi+1 xi



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Riemann sums contd

Riemann sum

S =n

i=1

f(i)(xi xi1)

where the i are taken in the intervals[xi1, xi].

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Riemann sums contd

Definite integral

x=b

x=a

f(x) dx = lim||P||0

n

i=1 f(i)(xixi1)



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Riemann sums contd

Evaluate using Riemann sums

x=tx=0

x2dx

Hint

12 + 22 + + n2 =2n3 + 3n2 + n

6
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Definite Integrals Sums Math 101