Aim what are riemann sums
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Aim: What are Riemann Sums?. Do Now:. Approximate the area under the curve y = 4 – x 2 for [-1, 1] using 4 inscribed rectangles. Devising a Formula. Using left endpoint to approximate area under the curve is. the more rectangles the better the approximation. lower sum.

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Aim: What are Riemann Sums?

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Aim what are riemann sums

Aim: What are Riemann Sums?

Do Now:

Approximate the area under the curve y = 4 – x2 for [-1, 1] using 4 inscribed rectangles.


Devising a formula

Devising a Formula

  • Using left endpoint to approximate area under the curve is

the more rectangles the better the approximation

lower sum

yn - 1

the exact area?

take it to the limit!

y1

yn - 1

y0

yn - 2

2

1

a

b

left endpoint formula


Right endpoint formula

Right Endpoint Formula

  • Using right endpoint to approximate area under the curve is

yn

upper sum

yn - 1

right endpoint formula

y1

y0

midpoint formula

a

b


Sigma notation

where i is the index of summation,

n is the upper limit of summation, and

1 is the lower limit of summation.

Sigma Notation

sum of terms

sigma

The sum of the first n terms of a sequence is represented by


Summation formulas

Summation Formulas


Riemann sums

Δx4

Δx5

Δx6

Δx1

Δx2

Δx3

x0

x1

x2

x3

x4

x5

x6

Riemann Sums

  • A function f is defined on a closed interval [a, b].

  • It may have both positive and negative values on the interval.

  • Does not need to be continuous.

Partition the interval into n subintervals not necessarily of equal length.

a = x0 < x1 < x2 < .. . < xn – 1< xn = b

a =

= b

Δxi = xi – xi – 1

- arbitrary/sample points for ith interval


Riemann sums1

Δx4

Δx5

Δx6

Δx1

Δx2

Δx3

Riemann Sums

  • Partition interval into n subintervals not necessarily of equal length.

x0

a =

x1

x2

x3

x4

x5

x6

= b

- arbitrary/sample points for ith interval

ci = xi


Riemann sums2

Δx6

Δx4

Δx2

Δx1

Riemann Sums

Δxi = xi – xi – 1

x6

x0

a =

= b


Definition of riemann sum

Definition of Riemann Sum

Let f be defined on the closed interval [a, b], and let Δ be a partition of [a, b] given by a = x0 < x1 < x2 < . . . . < xn – 1 < xn = b,

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

is called a Riemann sum for f for the partition Δ

largest subinterval – norm - ||Δ|| or |P|

equal subintervals – partition is regular

regular partition

general partition

converse not true


Model problem

Model Problem

Evaluate the Riemann Sum RP for

f(x) = (x + 1)(x – 2)(x – 4) = x3 – 5x2 + 2x + 8 on the interval [0, 5] using the Partition P with partition points 0 < 1.1 < 2 < 3.2 < 4 < 5 and corresponding sample points


Model problem1

Model Problem


Definition of definite integral

Definition of Definite Integral

If f is defined on the closed interval [a, b] and the limit

exists, the 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.

Definite integral is a number

Indefinite integral is a family of functions

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


Evaluating a definite integral as a limit

Evaluating a Definite Integral as a Limit

ci = xi


Evaluating a definite integral as a limit1

Evaluating a Definite Integral as a Limit

not the area

The Definite Integral as Area of Region

If f is continuous and nonnegative on the closed interval [a, b], then the area of the region bounded by the graph of f, the x-axis and the vertical lines x = a and x = b is given by


Properties of definite integrals

Properties of Definite Integrals


Areas of common geometric figures

Areas of Common Geometric Figures

Sketch & evaluate area region using geo. formulas.

= 8

A = lw


Model problems

Model Problems

=0


Model problem2

Model Problem


Model problem3

A2

A1

Total Area = -A1 + A2

Model Problem


Model problem4

A2

A1

Model Problem


Model problem5

Model Problem

take the limit n


Definition of riemann sum1

Definition of Riemann Sum


Definition of riemann sum2

Definition of Riemann Sum


Subintervals of un equal lengths

Subintervals of Unequal Lengths


Subintervals of un equal lengths1

Subintervals of Unequal Lengths


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