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Μάθημα: Ολοκληρωτικός Λογισμός > Ενότητα 1

Μάθημα 2: Approximation with Riemann sums

Αριστερά και δεξιά αθροίσματα Riemann

Τα εμβαδά κάτω από τις καμπύλες μπορούν να υπολογισθού με ορθογώνια. Τέτοιες εκτιμήσεις ονομάζονται αθροίσματα Riemann.

Ας υποθέσουμε ότι θέλουμε να βρούμε το εμβαδόν κάτω από αυτή την καμπύλη:

Μπορεί να αγωνιστούμε για να βρούμε το ακριβές εμβαδόν, αλλά μπορούμε να το προσεγγίσουμε χρησιμοποιώντας ορθογώνια:

Και η προσέγγισή μας γίνεται καλύτερη αν χρησιμοποιήσουμε περισσότερα ορθογώνια:

Αυτές οι προσεγγίσεις ονομάζονται Αθροίσματα Riemann, και είναι ένα θεμελιώδες εργαλείο για τον ολοκληρωτικό υπολογισμό. Ο στόχος μας, προς το παρόν, είναι να επικεντρωθούμε στην κατανόηση δύο τύπων αθροισμάτων Riemann: αριστερά αθροίσματα Riemann, και δεξιά αθροίσματα Riemann .

Αριστερά και δεξιά αθροίσματα Riemann

Για να φτιάξουμε ένα άθροισμα Riemann, πρέπει να επιλέξουμε πώς θα κάνουμε τα ορθογώνια μας. Μια πιθανή επιλογή είναι να κάνουμε τα ορθογώνια μας να αγγίξουν την καμπύλη με τις επάνω αριστερές γωνίες τους. Αυτό ονομάζεται αριστερό άθροισμα του Riemann.

Μια άλλη επιλογή είναι να κάνουμε τα ορθογώνια μας να αγγίξουν την καμπύλη με τις επάνω δεξιές γωνίες τους. Αυτό είναι ένα σωστό άθροισμα Riemann.

Καμία επιλογή δεν είναι καλύτερη από την άλλη.

Πρόβλημα 1

Τι είδους άθροισμα Riemann περιγράφεται από το διάγραμμα;

Riemann άθροισμα υποδιαιρέσης/τμήματα

Terms commonly mentioned when working with Riemann sums are "subdivisions" or "partitions." These refer to the number of parts we divided the

x

-interval into, in order to have the rectangles. Simply put, the number of subdivisions (or partitions) is the number of rectangles we use.

Subdivisions can be uniform, which means they are of equal length, or nonuniform.

Uniform subdivisions	Nonuniform subdivisions

Πρόβλημα 2

What is the correct description of the subdivisions in this Riemann sum.

Riemann sum problems with graphs

Imagine we're asked to approximate the area between

y = g (x)

and the

x

-axis from

x = 2

x = 6

And say we decide to use a left Riemann sum with four uniform subdivisions.

Notice: Each rectangle touches the curve at its top-left corner because we're using a left Riemann sum.

Adding up the areas of the rectangles, we get

20

units

^{2}

, which is an approximation for the area under the curve.

For this problem, we can just count the unit squares for each rectangle, but in general we'd need to multiply heights and widths to find area.

	First rectangle	Second rectangle	Third rectangle	Fourth rectangle
Width	$1$ ‍	$1$ ‍	$1$ ‍	$1$ ‍
Height	$3$ ‍	$7$ ‍	$6$ ‍	$4$ ‍
Area	$1 \cdot 3 = 3$ ‍	$1 \cdot 7 = 7$ ‍	$1 \cdot 6 = 6$ ‍	$1 \cdot 4 = 4$ ‍

Then, after finding the individual areas, we'd add them up to get

20

units

^{2}

Πρόβλημα 3

Approximate the area between $y = h (x)$ ‍ and the $x$ ‍-axis from $x = - 2$ ‍ to $x = 4$ ‍ using a right Riemann sum with three equal subdivisions.

Now let's do some approximations without the aide of graphs.

Imagine we're asked to approximate the area between the

x

-axis and the graph of

f

from

x = 1

x = 10

using a right Riemann sum with three equal subdivisions. To do that, we are given a table of values for

f


$x$ ‍	$1$ ‍	$4$ ‍	$7$ ‍	$10$ ‍
$f (x)$ ‍	$6$ ‍	$8$ ‍	$3$ ‍	$5$ ‍

A good first step is to figure out the width of each subdivision. The width of the entire area we are approximating is

10 - 1 = 9

units. If we're using three equal subdivisions, then the width of each rectangle is

9 \div 3 = 3

From there, we need to figure out the height of each rectangle. Our first rectangle sits on the interval

[1, 4]

. Since we are using a right Riemann sum, its top-right vertex should be on the curve where

x = 4

, so its

y

-value is

f (4) = 8

In a similar way we can find that the second rectangle, that sits on the interval

[4, 7]

, has its top-right vertex at

f (7) = 3

Our third (and last) rectangle has its top-right vertex at

f (10) = 5

Now all that remains is to crunch the numbers.

	First rectangle	Second rectangle	Third rectangle
Width	$3$ ‍	$3$ ‍	$3$ ‍
Height	$8$ ‍	$3$ ‍	$5$ ‍
Area	$3 \cdot 8 = 24$ ‍	$3 \cdot 3 = 9$ ‍	$3 \cdot 5 = 15$ ‍

Then, after finding the individual areas, we'd add them up to get our approximation:

48

units

^{2}

Πρόβλημα 4

Approximate the area between the $x$ ‍-axis and $y = g (x)$ ‍ from $x = 10$ ‍ to $x = 16$ ‍ using a left Riemann sum with three equal subdivisions.


$x$ ‍	$10$ ‍	$12$ ‍	$14$ ‍	$16$ ‍
$g (x)$ ‍	$5$ ‍	$1$ ‍	$7$ ‍	$7$ ‍

The approximate area is

units

^{2}

Now imagine we're asked to approximate the area between the

x

-axis and the graph of

f (x) = 2^{x}

from

x = - 3

x = 3

using a right Riemann sum with three equal subdivisions.

The entire interval

[- 3, 3]

6

units wide, so each of the three rectangles should be

6 \div 3 = 2

units wide.

The first rectangle sits on

[- 3, - 1]

, so its height is

f (- 1) = 2^{- 1} = 0.5

. Similarly, the height of the second rectangle is

f (1) = 2^{1} = 2

and the height of the third rectangle is

f (3) = 2^{3} = 8

	First rectangle	Second rectangle	Third rectangle
Width	$2$ ‍	$2$ ‍	$2$ ‍
Height	$0.5$ ‍	$2$ ‍	$8$ ‍
Area	$2 \cdot 0.5 = 1$ ‍	$2 \cdot 2 = 4$ ‍	$2 \cdot 8 = 16$ ‍

So our approximation is

21

units

^{2}

Πρόβλημα 5

Approximate the area between the $x$ ‍-axis and $h (x) = \frac{3}{x}$ ‍ from $x = 0$ ‍ to $x = 1.5$ ‍ using a right Riemann sum with $3$ ‍ equal subdivisions.

The approximate area is

units

^{2}

Want more practice? Try this exercise.

Riemann sums sometimes overestimate and other times underestimate

Riemann sums are approximations of the area under a curve, so they will almost always be slightly more than the actual area (an overestimation) or slightly less than the actual area (an underestimation).

Πρόβλημα 6

Is this Riemann sum an overestimation or underestimation of the actual area?

Πρόβλημα 7

Consider the left and right Riemann sums that would approximate the area under

y = g (x)

between

x = 2

and

x = 8

Are the approximations overestimations or underestimations? Fill in the blanks.

The left Riemann sum would be entirely

the curve, so it would be an

The right Riemann sum would be entirely

the curve, so it would be an

Πρόβλημα 8

The continuous function

g

is graphed.

We're interested in the area under the curve between

x = - 7

and

x = 7

, and we're considering using Riemann sums to approximate it.

Order the areas from least (on top) to greatest (on bottom).

Πρόβλημα 9

This table gives select values of the continuous and increasing function

g

$x$ ‍	$- 2$ ‍	$3$ ‍	$8$ ‍	$13$ ‍	$18$ ‍
$g (x)$ ‍	$13$ ‍	$19$ ‍	$28$ ‍	$31$ ‍	$41$ ‍

We're interested in the area under the curve between

x = - 2

and

x = 18

, and we're considering using left and right Riemann sums, each with four equal subdivisions, to approximate it.

Order the areas from least (on top) to greatest (on bottom).

Want more practice? Try this exercise.

Notice: Whether a Riemann sum is an overestimation or an underestimation depends on whether the function is increasing or decreasing on the interval, and on whether it's a left or a right Riemann sum.

Key points to remember

Approximating area under a curve with rectangles

The first thing you should think of when you hear the words "Riemann sum" is that you're using rectangles to estimate the area under a curve. In your mind, you should envision something like this:

Better approximation with more subdivisions

In general, the more subdivisions (i.e. rectangles) we use to approximate an area, the better the approximation.

Left vs. right Riemann sums

Try not to mix them up. A left Riemann sum uses rectangles whose top-left vertices are on the curve. A right Riemann sum uses rectangles whose top-right vertices are on the curve.

Left Riemann sum‍	Right Riemann sum

Overestimation and underestimation

When using Riemann sums, sometimes we get an overestimation and other times we get an underestimation. It's good to be able to reason about whether a particular Riemann sum is overestimating or underestimating.

In general, if the function is always increasing or always decreasing on an interval, we can tell whether the Riemann sum approximation will be an overestimation or underestimation based on whether it's a left or a right Riemann sum.

Direction	Left Riemann sum	Right Riemann sum
Increasing	Underestimation	Overestimation
Decreasing	Overestimation	Underestimation

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