Question

solve the following using midpoint rule and simpson rule..
integral of (x/(1+x^2)), [0,2], n=10

Answer 1

Would be happy to help, but invite you to be involved as much as you possibly can be.

In this problem we're aiming to find the area between the graph of the given function on the interval that begins with a and ends with b. We'll divide up that area into n vertical strips.

Then the width of each vertical strip of area is

\[\Delta x=\frac{ b-a }{ n }\]
Would you calculate this width, please?

Answer 2

Sure. delta x= 2-0/10=1/5

Answer 3

That's correct. Good for you. One suggestion, however: Please enclose that x-2 inside parentheses: (2-0)/10.

Next, we need to find n+1 x-values at which to evaluate f(x).
The first one will be a = 0, and the second b = 2.

What are the 9 intermediate values?

Answer 4

Is it starts with 0.1, 0.3...?

Answer 5

I'm very sorry. I meant that the first x-value would be 0 and the LAST would be 2.

If the first x-value is 0, then our set of x-values starts out as 0, 0.2, 0.4, ... , and ends with with ... ??

Answer 6

Remember, delta x is 0.2.

Answer 7

set of x is 0,0.2,0.4,...,2
intermediate value is 0.1,0.3,0.5,....1.9 right?

Answer 8

There are n+1 such x- values, meaning 11 x-values.
They are, as you said above, 0,0.2,0.4,...,2.

(Save the intermediate values for the "midpoint" approach, which we'll tackle later. For now, we're focusing on the use of Simpson's Rule.

Answer 9

lala: Have you any references available, e. g., your course textbook, access to Internet searches, or the like? If so, could you please look up the formula for Simpson's Rule, if you don't know it already?

Answer 10

Formula: h/3(y0+4y1+y2)

Answer 11

that formula will be our starting point, towards approximating the area under the curve between x=0 and x=2.

Your h/3(y0+4y1+y2) is all right as a starting point. Again, please enclose h/3 within parentheses so that there's no ambiguity about h/3 being a fraction.

Since n=10, you will have n+1, or 11, x-values and 11 corresponding y values.

How do you normally go about evaluating a function such as f(x) = x / (1+x^2) at various x-values?

Answer 12

Starting with your (h/3(y0+4y1+y2+ ..... ), let's write Simpson's Rule more formally:

Answer 13

\[A=\frac{ \Delta x}{ 3 }(1*f(x _{0}) + 4*f(x _{1}) + 2*f(x _{2}) + ,,, +4*f(x _{9})+1*f(x _{10}))\]

Answer 14

I'm assuming you've seen this format before. Right?

Answer 15

Yes!

Answer 16

And we already know that delta x is 1/5, or 0.2, and that the first x-value is zero, and so on.
So, what is the value of the function at x0 = 0?

Answer 17

Is it 0?

Answer 18

Yes. Basically you'll need to create a table with those 11 x-values in the left column and the values of the function x/(1+x^2) in the second column. I asked you before how you 'd go about calculating those function values.

Answer 19

Lala, I have begun such a table, using Excel on my computer. Are you able to finish this table, that is, are you able to calculate the remaining function values?

Answer 20

yes!

Answer 21

OK, then please go back to to the formula I typed in earlier.

Area = A = [(delta x)/3]*(1*f(0 + 4*f(0.2) + ................ ).

Answer 22

could you describe in words what you have to do to finish this problem?

Answer 23

substitute values of f(x) from x0 to x11 into the formula

Answer 24

Let me re-word that a bit:
"substitute the 11 x-values into the formula f(x), one by one.

Answer 25

Right. And then what?

Answer 26

Yes! Then, calculate the answer by multiply (delta x/3) with the total of f(x)

Answer 27

Hint: remember those coefficients, {1,4,2,4,2,4, ... 4,1}?

Not "total of f(x)," but "the sum of the products 1*f(0),4*f(0.2), 2*f(0.4), and so on."

Answer 28

Yes, that's correct^_^

Answer 29

Organizing all this info in a table should be helpful. I've come up with the following table and ask you whether or not you feel comfortable completing it:

Answer 30

Can you view this table? If so, what's the next thing you'd do to complete this table?

Answer 31

yes. now I have all the coeff*functions value. I am going to total them up

Answer 32

that's right: you'll add up those 11 products. Then y ou'll multiply this sum by (delta x)/3, remembering that delta x is 0.2. OK?

Answer 33

Yes. I did it and got the answer^_^

Answer 34

Note that in using Simpson's Rule, n must be EVEN, and n+1 must be ODD. Our work here obeys those guidelines.

Cool!

think: How would this work be different if we were to use midpoints instead of subinterval endpoints as we just did with Simpson's Rule?

Answer 35

One hint: We won't need coefficients such as 1, 4, 2, 4, ...., 4, 1.

Answer 36

Lala?

Answer 37

Into how many strips are we dividing the area under the graph of f(x) between 0 and 2?

Answer 38

9?

Answer 39

My mistake. It's 12

Answer 40

fOR Simpston's Rule it'd be 10 strips of width 0.2, and we'd need 10+1, or 11, y-values.

but when using the Midpoint Rule, we need just 10 strips, each of width 0.2. We calculate the height of each strip and multiply each height by 0.2. Why?

Answer 41

I thought the width is 0.1?

Answer 42

No, the width is (2-0)/10, as before. When you typed in 0.1, you were referring to the midpoint of the first strip. The first strip begins at 0.0 and ends at 0.2, since the strip width is 0.2; the midpoint of the interval [0,0.2] is indeed 0.1.

Answer 43

Please write out the 10 midpoints.
Start with 0.1.
For each new midpoint, simply add the width 0.2 to the previous midpoints.

The 10 midpoints are then : ?

Answer 44

0.1,0.3,0.5,0.7,0.9,1.1,1.3,1.5,1.7,1.9

Answer 45

midpoint formula: (delta x/2)(f(0.1)+f(0.3)+......+f(1.9). Am I correct?

Answer 46

Looks good. So, you'll be calculating 10 (not 11) function values, multiplying each by the strip width, and then adding up the 10 products. That's it. Less work than Simpson's Rule, but usually not as accurate as simpson's Rule.
You OK with this? Any questions about the processes we've gone thru?

Answer 47

Nope! Thank you so much^_^

Answer 48

It's been a great pleasure to work with you, seeing that you already know a great deal about these topics! Bye, Lala. MM

Answer 49

Bye Bye