I need some help estimating the integral from -3 to 3 on this graph: https://instruct.math.lsa.umich.edu/webwork2_course_files/ma115-173-w11/tmp/gif/mcdotreb-4414-setChap5Sec3prob7image1.png
How do I estimate the integral with only a graph? I have tried rectangles, but I am not getting the right answer. Please Help!

- anonymous

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- anonymous

the integral from -3 to 3 on that graph is the area of that graph from -3 to 3

- anonymous

So I need to find the area under the curve between the curve and the x-axis. But how, when there is not concrete numbers on the graph to work with?

- anonymous

You need to be careful: the 'area' under the curve when it is in negative y-territory is negative.

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- anonymous

Right, and I add the absolute value of both pieces because they both are in negative territory.

- anonymous

Have you heard of Simpson's Rule?

- anonymous

No

- anonymous

Okay, so it sounds like they only want you estimating with rectangles to get you used to the Riemann sum (which is essentially what this kind of integration is).

- anonymous

We don't have any other instructions in this section of my math book except to count squares, do the left-hand and right-hand estimation, and use the intergral (but not the antidrevative).

- anonymous

I would:
(1) estimate the area under the curve *above* the x-axis
(2) estimate the area under the curve *below* the x-axis
Subtract (2) from (1): that is, \[\int\limits_{}{} \approx A_1-A_2\]

- anonymous

To estimate A1, I tried to count the rectangles, but I could't make it work

- anonymous

Why couldn't you make it work?

- anonymous

There is one full square with and area of one, but then there are two peices of a square that can't be made to fit together to make another square. So the are for -3 to 0 is 1. something, I think.

- anonymous

I get roughly 4.25 above the x-axis. This is an approximation.

- anonymous

You use all the squares the graph touches, even if it doesn't make a square?

- anonymous

Yeah, you don't have to be perfect. There isn't enough information. That's why I asked if you'd heard of Simpson's Rule because you could get this done and dusted in a couple of minutes.

- anonymous

What's simpsons rule (the gist of it...I have looked it up on Wikipedia just a moment ago)

- anonymous

I just estimate what fraction of a square is under the curve if the graph cuts it. That's all.

- anonymous

\[\int\limits_{a}^{b}f(x)dx \approx \frac{b-a}{6}\left[ f(a)+4f(\frac{a+b}{2})+f(b) \right]\]

- anonymous

So four square that get mostly touched by the curve, plus the little bit in the corner?

- anonymous

This is the simpson's rule? Thanks

- anonymous

i got 3.5

- anonymous

what was your method?

- anonymous

dindatc probably took his/her estimations with more refinement.

- anonymous

I think the main point of the exercise is to teach you the geometrical motivation for what is going on, and to realise the difference between 'integral' and 'area' - i.e. the integral in negative regions is negative, whereas area is always positive. If you haven't used Simpson's Rule in class, stick to rectangular estimation.

- anonymous

sorry i calculate it mistakenly, i use the (2/3) * base * height
and i got 5

- anonymous

in the -x area, I got approx. maybe 2.5, looking at the graph. What is wrong with this estimation?

- anonymous

I know it is wrong, but I don't see how to do the rectangle estimation, I suppose. This problem doesn't come with an equation.

- anonymous

I get, overall, as a rough estimation, -3.25 as the integral value from x= -4 to 4.

- anonymous

##### 1 Attachment

- anonymous

I'm going to estimate using Simpson's Rule. You can use it to compare with your rectangular estimation.

- anonymous

okay

- anonymous

thanks dindatc. I am looking at the graph right now...where do you get 2.5 from? Below the graph, I see where you got 4 from, but not 2/3 or 2.5

- anonymous

i am really interested in seeing how simpson's rule comes out

- anonymous

2/3 is from the formula
the formula is (2/3) x base x height
which 2.5? both area have 2.5

- anonymous

Yes, where do you estimate 2.5 for both

- anonymous

What rule are you using...I just understood the base part now!

- anonymous

the left side has a base of 2.5, and the right has a base of 4

- anonymous

on the left one, the graph intersect x-axis on x=-1 and x = -3.5(approximately) so the base would be -1-(-3.5) = 2.5
on the right one, 2.5 is the height of that shape, i marked it with green line

- anonymous

okay, I think I am starting to see this...how did you know how to do this? Is it a certain rule?

- anonymous

my teacher taught me that formula, he said it's a faster way to estimate the area of a parabola

- anonymous

(2/3)*base*height?

- anonymous

yes

- anonymous

if you could find the equation of that graph, i think you could get a better estimation

- anonymous

I wonder why they didn't teach us that...would save much frustration!

- anonymous

you would approximate that in the -x area, the height is 1 or more than that?

- anonymous

i approximate the height is 1

- anonymous

I think it is more, with that small piece at the top of the box. How did you get one?

- anonymous

##### 1 Attachment

- anonymous

I had to finish something else.

- anonymous

i chose 1 to make it easier to calculate

- anonymous

I estimated -3.25 using rough estimation with rectangles (eyeballing) and this calculation doesn't disagree too much, given it's again, an estimation.

- anonymous

but the actual height is more than one, so the differences of the two areas will be smaller, therefore the answer will be smaller than 5

- anonymous

for some reason, the computer won't take either method's answers. :(

- anonymous

I understand what you are doing dindatc, and I looked at your attachment, lokisan. I think I understand it, based on the rules...

- anonymous

@lokisan why you also calculate the A0 and A3?

- anonymous

Sorry, ilovemath, I can't offer any more answers. This is the problem with online submission...I hate it.

- anonymous

Because you have to find the area under the curve, don't you?

- anonymous

but isn't interval from -3 to 3?

- anonymous

Yes, from the interval -3 to 3. I have used up all my tries but nothing has worked.

- anonymous

Oh, yes...ummm...I'm tired...

- anonymous

You can still take the A_2 estimation. Just have to calculate the estimation fro -3 to 0.

- anonymous

Thanks so much for helping me, Lokisan and dindatc! I will continue to work on it, and bring it to the mathlab to see what they say. I wish all we had was written homework.

- anonymous

I mean, -1 to 3 is okay. Have to sort out -3 to -1.

- anonymous

You're welcome.

- anonymous

if it's only A1 and A2, the area is 14/3

- anonymous

you're welcome

- anonymous

peace

- anonymous

I get -4.9333...

- anonymous

Area from -3 to -1 is approx 1.73, and the area from -1 to 3 is approx -6.67

- anonymous

yep, our answer is almost the same

- anonymous

Phew...

- anonymous

no dice. computer will not accept the approximation, which I believe will be 8.4? I don't know what to tell you...

- anonymous

Where'd you get 8.4?

- anonymous

The integral is 1.73-6.67=-4.94

- anonymous

do you have to add them, since area must be positive?

- anonymous

no, the area under the x-axis is negative

- anonymous

ohhhhhhhhhhhhhhhhhh. estimation proved correct. sorry

- anonymous

If you're looking for area, then you add the magnitudes, but if it's the integral, that part *under* the x-axis will be negative.

- anonymous

seeee

- anonymous

so, the integral can be a negative. i use the estimations and subtract them because they are on different sides of the x axis

- anonymous

All the 'area' under the x-axis is negative. This comes from the definition of the integral You have to remember that it's taking function values for the 'height' of the rectangle. When the curve is UNDER the x-axis, the function values are NEGATIVE, so the height is NEGATIVE which means \[ \delta x \times f(x)\]will be negative also

- anonymous

okay

- anonymous

but area will be the absolute value of this, right?

- anonymous

Yes.

- anonymous

the result -4.9 means the area is 4.9 because an area will always positive

- anonymous

If it asks for area, that is positive. So in that instance, you'd have to be careful with a function and not just integrate from one end to the other if part of the function hits negative territory. You'd have to integrate up to the point where the function cuts the x-axis, record the value, keep going during the interval where the integral is negative, take note of that value, and then continue where it's positive.

- anonymous

There's a difference between *integral* and *are*.

- anonymous

*area*.

- anonymous

thanks

- anonymous

np

- nikvist

\[f(x)\approx\frac{1}{7}(x-3)(x+1)(x+3.5)\quad,\quad\int\limits_{-3}^{3}f(x)dx\approx -5\frac{1}{7}\]

- anonymous

so this what my equation would look like if I wanted to create one to make estimation easier?

- anonymous

I have to leave now...I'm *very* tired. You got your answer in the end.

- anonymous

yes, and thanks very much!

- anonymous

you're welcome.

- anonymous

even if the graph splits through the first box, we still use it to total 7 boxes for the 1/7 (meaning base)?

- anonymous

wait...this doesn't give you the right graph...

- anonymous

oops. nevermind

- anonymous

nope. still wrong answer with the equation. Thanks though

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