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?

Do Now:

Approximate the area under the curve y = 4 – x2 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

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
• Using right endpoint to approximate area under the curve is

yn

upper sum

yn - 1

right endpoint formula

y1

y0

midpoint formula

a

b

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

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

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

Δx6

Δx4

Δx2

Δx1

Riemann Sums

Δxi = xi – xi – 1

x6

x0

a =

= b

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

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

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

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

Areas of Common Geometric Figures

Sketch & evaluate area region using geo. formulas.

= 8

A = lw

A2

A1

Total Area = -A1 + A2

Model Problem
Model Problem

take the limit n