AP CALCULUS AB

Chapter 5:The Definite Integral

Section 5.2:Definite Integrals

What you’ll learn about Riemann Sums The Definite Integral Computing Definite Integrals on a

Calculator Integrability

… and whyThe definite integral is the basis of integral

calculus, just as the derivative is the basis of differential calculus.

Sigma Notation

1 2 3 11

...n

k n nk

a a a a a a

Section5.2 – Definite Integrals Definition of a Riemann Sum

f is defined on the closed interval [a, b], and is a partition of [a, b] given by

where is the length of the ith subinterval.If ci is any point in the ith subinterval, then the sum

is called a Riemann Sum of f for the partition .

bxxxxxa nn 1210 ...

ix

iii

n

iii xcxxcf

1

1

a bPartitions do not have tobe of equal width

If the are of equal width,then the partition is regular and

lsubintervalargest

theofwidth norm

n

abx

The Definite Integral as a Limit of Riemann Sums

-1

0

Let be a function defined on a closed interval [ , ]. For any partition

of [ , ], let the numbers be chosen arbitrarily in the subinterval [ , ].

If there exists a number such that lim

k k k

P

f a b P

a b c x x

I 1

( )

no matter how and the 's are chosen, then is on [ , ] and

is the of over [ , ].

n

k kk

k

f c x I

P c f a b

I f a b

integrable

definite integral

The Existence of Definite Integrals All continuous functions are integrable. That is, if a function is

continuous on an interval [ , ], then its definite integral over

[ , ] exists.

f

a b

a b

The Definite Integral of a Continuous Function on [a,b]

1

Let be continuous on [ , ], and let [ , ] be partitioned into subintervals

of equal length ( - ) / . Then the definite integral of over [ , ] is

given by lim ( ) , where each is chon

k kn k

f a b a b n

x b a n f a b

f c x c

th

sen arbitrarily in the

subinterval.k

The Definite Integral

( )b

a f x dx

Section 5.2 – Definite Integrals If f is defined on the closed interval [a, b] and the

limit

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

The limit is called the definite integral of f from a to b. The number a is the lower limit of integration, and the number b is the upper limit of integration.

n

iii xcf

10

lim

.lim1

0

b

a

n

iii dxxfxcf

The function is the integrand

x is the variable of integration

Example Using the Notation

th

2

1

The interval [-2,4] is partitioned into subintervals of equal length 6 / .

Let denote the midpoint of the subinterval. Express the limit

lim 3 2 5 as an integral.

k

n

k kn k

n x n

m k

m m x

2 4 2

21

lim 3 2 5 3 2 5n

k kn km m x x x dx

Section 5.2 – Definite Integrals Theorem: If y=f(x) is

nonnegative and integrableover a closed interval [a, b],then the area under the curve

y=f(x) from a to b is the

integral of f from a to b,

If f(x)< 0, from a to b (curve is under the x-axis),

then

a b

.b

adxxfArea

b

adxxfArea .

Area Under a Curve (as a Definite Integral) If ( ) is nonnegative and integrable over a closed interval [ , ],

then the area under the curve ( ) from to is the

, ( ) .b

a

y f x a b

y f x a b

A f x dx

integral

of from to f a b

Area

Area= ( ) when ( ) 0.

( ) area above the -axis area below the -axis .

b

a

b

a

f x dx f x

f x dx x x

The Integral of a Constant If ( ) , where is a constant, on the interval [ , ], then

( ) ( ) b b

a a

f x c c a b

f x dx cdx c b a

Section 5.2 – Definite Integrals To find Total Area Numerically (on the

calculator)

To find the area between the graph of y=f(x) and the x-axis over the interval

[a, b] numerically, evaluate:On the TI-89:

nInt (|f(x)|, x, a, b)On the TI-83 or 84:

fnInt (|f(x)|, x, a, b) Note: use abs under Math|Num for absolute value

Example Using NINT (FnInt)

2

-1Evaluate numerically. sinx xdx

NINT( sin , , -1,2) 2.04x x x

Example Using NINT (FnInt)

2

-1Evaluate numerically. sinx xdx

NINT( sin , , -1,2) 2.04x x x

Discontinuous FunctionsThe Reimann Sum process guarantees that all functions that are continuous are integrable. However, discontinuous functions may or may not be integrable.

Bounded Functions: These are functions with a top and bottom, and a finite number of discontinuities on an interval [a,b]. In essence, a RAM is possible, so the integral exists, even if it must be calculated in pieces. A good example from the Finney book is f(x) = |x|/x.

Discontinuous FunctionsAn example of a discontinuous function (badly discontinuous), which is also known as a non-compact function, is given also:

This function is 1 when x is rational, zero when x is irrational. On any interval, there are an infinite number of rational and irrational values.