Calculus:Calculus:

Riemann Sums & Riemann Sums & Definite IntegralsDefinite Integrals

Section 6.3Section 6.3

Finding Area with Riemann SumsFinding Area with Riemann Sums

• For convenienceFor convenience, , the area of a the area of a partition is often partition is often divided into divided into subintervals with subintervals with equal width – in equal width – in other words, the other words, the rectangles all have rectangles all have the same width. the same width.

4

3

2

1

2

f x = x2

Subintervals with equal width

Finding Area with Riemann SumsFinding Area with Riemann Sums

• It is possible to divide a region into It is possible to divide a region into different sized rectangles based on different sized rectangles based on an algorithm or rulean algorithm or rule

6

4

2

5 10 15

Unequal Subintervals

Finding Area with Riemann SumsFinding Area with Riemann Sums

• It is also possible to make rectangles of It is also possible to make rectangles of whatever width you want where the width whatever width you want where the width and/or places where to take the height and/or places where to take the height does not follow any particular pattern.does not follow any particular pattern.

Notice that the subintervals don’t seem to have a pattern. They don’t have to be any specific width or follow any particular pattern.

Also notice that the height can be taken anywhere on each subinterval – not only at endpoints or midpoints!

4

2

Riemann SumsRiemann SumsDefinitionDefinition: :

Let Let f f be defined on the closed interval [a,b], and let be defined on the closed interval [a,b], and let be a partition [a,b] given bybe a partition [a,b] given by

a = xa = x00 < x < x11 < x < x22 < … < x < … < xn-1n-1 < x < xnn = b = b

where where xxii is the width of the is the width of the ith ith subinterval [xsubinterval [xi-1i-1, x, xii]. ]. If c is any point in the If c is any point in the ithith subinterval, then the sum subinterval, then the sum

i-1 i n1

( ) x c x n

i ii

f c x

is called a is called a Riemann sumRiemann sum of of ff for the partition for the partition ..

New Notation for New Notation for xx

• When the partitions (boundaries that tell When the partitions (boundaries that tell you where to find the area) are divided into you where to find the area) are divided into subintervals with different widths, the width subintervals with different widths, the width of the largest subinterval of a partition is the of the largest subinterval of a partition is the normnorm of the partition and is denoted by || of the partition and is denoted by ||||||

• If every subinterval is of equal width, the If every subinterval is of equal width, the partition is partition is regularregular and the norm is denoted and the norm is denoted byby

|| ||||= ||= x = x =

• The number of subintervals in a partition The number of subintervals in a partition approaches infinity as the norm of the approaches infinity as the norm of the partition approaches 0. In other words, ||partition approaches 0. In other words, |||| || 0 implies that n 0 implies that n

b - an

Definite IntegralsDefinite IntegralsIf If f f is defined on the closed interval [a,b] and is defined on the closed interval [a,b] and the limitthe limit

01

lim ( )n

i ii

f c x

exists, then f is exists, then f is integrableintegrable on [a,b] and the on [a,b] and the limit is denoted bylimit is denoted by

The limit is called the The limit is called the definite integraldefinite integral of of ff from from aa to to bb. .

The number The number aa is the is the lower limit of integrationlower limit of integration and the and the number number bb is the is the upper limit of integrationupper limit of integration..

01

( )lim ( )n

i ii

a

bf c f x dxx

Definite Integrals vs. Indefinite Definite Integrals vs. Indefinite IntegralsIntegrals

A A definite integraldefinite integral is is numbernumber..

An An indefinite integralindefinite integral is a is a family of functionsfamily of functions. .

They may look a lot alike, however, They may look a lot alike, however, • definite integrals have limits of integrationdefinite integrals have limits of integration while the while the • indefinite integrals have not limits of indefinite integrals have not limits of integration.integration. ( )

b

af x dx ( )f x dx

Definite Definite IntegralIntegral

Indefinite Indefinite IntegralIntegral

Theorem 6.4Theorem 6.4 Continuity Implies Continuity Implies IntegrabilityIntegrability

If a function If a function ff is is continuous on the closed continuous on the closed interval [interval [a,ba,b], then ], then ff is is integrable on [integrable on [a,ba,b].].

Is the converse of this statement true? Why or why not?

Theorem 6.5Theorem 6.5 The Definite Integral as the Area of a The Definite Integral as the Area of a RegionRegionIf If ff is is continuouscontinuous and and nonnegativenonnegative on the closed on the closed interval [a,b], then the interval [a,b], then the area of the region bounded area of the region bounded by the by the graph of graph of ff , the , the x-x-axisaxis, and the vertical lines , and the vertical lines x = ax = a and and x = bx = b is given by is given by

( )b

af x dxArea =Area =

5

4

3

2

1

-1

-2 2

f(x) = -0.61x4 + -1.14x3 + 1.75x2 + 2.71x + 2.46(x,f(B))

(x,f(A))

1A B

Let’s try this out…

1. Sketch the region

2. Find the area indicated by the integral.

3

14 dx

8

6

4

2

-2

5Area

= (base)(height)

= (2)(4) = 8 un 2

1. Sketch the region

2. Find the area indicated by the integral.

3

0(x+2) dx

8

6

4

2

-2

5

Area of a trapezoid

= .5(width)(base1+ base2)

= (.5)(3)(2+5) = 10.5 un 2

Width =3

base1=2

base2 =5

1. Sketch the region

2. Find the area indicated by the integral.

2 2

24 x dx

2

Area of a semicircle

= .5( r2)

= (.5)()(22)

= 2 un 2

Properties of Definite IntegralsProperties of Definite Integrals

• If If f f is defined at x = a, then we define is defined at x = a, then we define

( ) 0a

af x dx

(x) (x) b

b

a

af dx f dx

So, So,

If If f f is integrable on [a,b], then we define is integrable on [a,b], then we define

So, So,

sin x dx

0

3( 2) x dx

Additive Interval Additive Interval PropertyProperty

(x) (x) (x) a a c

bcbf dx f dx f dx

If If f f is integrable on is integrable on three closed three closed intervals intervals determined by a, b, determined by a, b, and c, then and c, then

Theorem 6.6Theorem 6.6

4

3

2

1

-1

-2

-3

-8 -6 -4 -2 2

0 1 2 3 4 5-1-2-3-4-5

C

To morph f, drag a,b,c,d,e.To rescale sliders, drag red tick mark at 1.

n = 1

f(x) dx = 7.0921.427

Dbl-click n to changeOR select & type +/-

-1.498

f(x) = -0.61x4 + -1.14x3 + 0.75x2 + 2.71x + 2.46

Show RS-LS Info.

n+10 nonstop, n<80

n+1 nonstop, n<80

Hide Integral Shading

Show Trapezoids

Show Left Rects.

Show Right Rects.

Show Mdpt. Rects.(x,f(B))

(x,f(A))

1A B

ab

cd

e

M7

N7

Properties of Properties of Definite Definite IntegralsIntegrals

(x) (x) b b

a akf dx k f dx

If If f and g f and g are integrable on [are integrable on [a,ba,b] and ] and kk is is a constant, then the functions of a constant, then the functions of kf kf and and f f g g are integrable on [ are integrable on [a,ba,b], and], and

Theorem6.7Theorem6.7

[ (x) ( )] (x) g(x) b b b

a a af g x dx f dx dx

Evaluating a definite integral…Evaluating a definite integral…

The Fundamental Theorem of Calculus

If f is continuous at every point of , and if

F is any antiderivative of f on , then

,a b

b

af x dx F b F a

,a b