5.2B Limits of Riemann 5.2B Limits of Riemann Sums and the Definite Sums and the Definite

IntegralIntegralToday, students will connect the limit of a Riemann Sum

to the Definite Integral, and learn the meaning and notation for the Definite Integral.

Earlier, we learned this…Earlier, we learned this…

…but we didn’t have much time to talk about what it means

What is that thing on the right-hand side?

x a x bSince the limit of the Riemann Sum gives the area between a function and the x-axis between and

it represents the same quantity as the definite integral:

Now, we can see that the “dx” represents a change in x, but as n has become infinitely large, Δx has become infinitesimally small. In other words, as , n 0x

2

82

5

Using the function write sigma notation to

approximate using:

a. Two right endpoint rectangles.

b. Nine right endpoint rectangles.

c. Twenty-four right endpoint rectangles.

d. Explain wh

f x x

x dx

y sigma notation requires that the rectangles have equal widths.

e. Do the right endpoint rectangles over or under approximate the actual area?

1

Remember the relationship between a

Riemann Sum and a Definite Integral:

limb n

kn

ka

I f x dx f c x

1

limb n

kn

ka

I f x dx f c x

Expanding sums:

100

1

2k

k

Summation properties and formulas (you are not being held responsible to know these )

ka k ai ii

n

i

n

11

( )a b a bi i ii

n

i

n

ii

n

11 1

c cni

n

1

in n

i

n

( )1

21

in n n

i

n2

1

1 2 1

6

( )( ) i

n n

i

n3

2 2

1

1

4

( )

constant multiple addition/subtraction

y x 2

n Find the right rectangular sum for the area under

on the interval [5,8] using n rectangles. Then, find the limit as

.

This is the limit of the Riemann Sum asWhat did we just find?

n

Using NINT (or on Using NINT (or on youryour calculator, calculator, fnInt)fnInt)

2

Find the area enclosed between the axis and the graph

of from 5 to 8.

x

y x x x

Method 1: press Math 9 and enter the following parameters:

Method 2: graph the function and use 2nd-Calc 7

Partner ProblemPartner Problem

The Relationship of the The Relationship of the Integral to AreaIntegral to Area

Signed AreaSigned Area

4

2

-2

1 2

Consider the graph to the left. If this is the graph of the velocity of an object moving along a horizontal line over time, what does the area between the graph and the t-axis mean?

The integral is defined as the “signed” area between a graph and the horizontal axis.

Area above the horizontal axis is defined as positively signed area. What does this area represent in terms of the movement of the particle? Area below the horizontal axis is defined as negatively signed area. What does this area represent in terms of the movement of the particle?

v t

t

2

1

1 2If an object moves with velocity ( ) on the interval , ,

then represents ____________________t

t

v t t t

v t dt

More on AreaMore on Area

This does not mean that area means anything different than it did in Geometry:

The Integral in relation to Geometric area:

More practice with signed areas:More practice with signed areas:

3

3

12

1

Use the graph of the integrand and Geometric areas to

evaluate the integrals:

1. 2 3

2. 1 1

x dx

x dx

Writing an integral to model a Writing an integral to model a situation:situation:

2

Beginning at a speed of 15 ft/sec., a car maintains a constant acceleration of

10 ft/sec for 10 seconds. Express the velocity of the car as a definite

integral and then evaluate the integral using Theorem 2.

2

Beginning at a standstill, a car maintains a constant acceleration of

10 ft/sec for 10 seconds. Express the velocity of the car as a definite

integral and then evaluate the integral using Theorem 2.

ExplorationExploration

AssignmentAssignment

5.2B: p. 282: (p. 282: 16,26,30,32,33,36, 41-46)