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#1 2012-04-25 07:04:08

amberzak
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Calculus - Area of a curve

Hi all. I am onto Calculus now. Thing is, I have only done 2 lessons and never studied it before. I'm getting on alright, but I have a really hard question (well, hard to me). It's an extra question, so if I don't do it it isn't a big deal, but I am trying to figure it out.

http://dl.dropbox.com/u/1023145/Screen%20Shot%202012-04-24%20at%2022.04.44.png

This (if it works) is a roughly drawn picture. It says:

The figure shows the curve with equation y=5+2x-x^2 and the line with equation y=2. The curve and the line intersect at the points A and B.

a) Find the x-coordinates of A and B.

The shaded region R is bounded by the curve and the line.

b) Find the area of R.

Can someone please take me through step by step how you find the x-Coordinates of A and B? I do have the answers by the way, just don;t know how to get to it. Our teacher gives us the final answers so we can check we have it right, and he marks us on our working.

The thing that's confusing me the most is that the equation isn't in the normal format, and so the x^2 value is negative, which I always thought meant an imaginary number, so I think I've missed something.

As I say, my teacher did say he doesn't mind if I don't do this question, as I have never done calculus before, but I am now intrigued as to how it's done.

Thanks all.


Don't think outside the box. Think there is no box

#2 2012-04-25 07:07:17

bobbym
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Re: Calculus - Area of a curve

Hi amberzak;

For the first problem. Since you know y = 2 then just plug it in to get the points of intersection.

a)



Can you solve that or do you need help?


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

#3 2012-04-25 07:29:37

amberzak
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Re: Calculus - Area of a curve

I don't want to sound dumb, but could you just put the solving up? I think I have it, but I just need to double check I've done it right. (My biggest problem is confidence)


Don't think outside the box. Think there is no box

#4 2012-04-25 07:40:17

bobbym
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Re: Calculus - Area of a curve

Hi;



Subtract 2 from both sides and turn it around.



Multiply everything by -1



Can you factor that?


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

#5 2012-04-25 07:50:01

amberzak
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Re: Calculus - Area of a curve

Thanks. I didn't do the multiply everything by -1.

So the factoring would be:
(x-3)(x+1) and the points at A and B would be 3 and -1. Is that right?


Don't think outside the box. Think there is no box

#6 2012-04-25 07:58:01

bobbym
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Re: Calculus - Area of a curve

Correct! Very good!


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

#7 2012-04-25 08:12:00

amberzak
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Re: Calculus - Area of a curve

Okay, so next part of the question. (I can't do the notation on here so I'm just going to write it without the correct notation).

Intergrating 5+2x-x^2.
=5x+2x^2 - x^3
        ------  ------
          2         3

Then supplement the two coordinates in for x
= (5x3 + 2x3^2 - x^3  ) - (5x-1 + 2x-1^2 - -1^3)
              -------   ------                 --------     -----
                 2         3                         2            3


and that's where I get a bit unstuck. The first part is easy enough:

=15 +9 -3) - (this is the bit i am sure I have wrong)


Don't think outside the box. Think there is no box

#8 2012-04-25 08:22:01

bob bundy
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Re: Calculus - Area of a curve

hi amberzak,

The area you have found is for all of the bit below the curve down to the  x axis.

The region R is less than that by one rectangle.

Once you subtract that, you should get the required answer.

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

#9 2012-04-25 08:27:28

bob bundy
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Re: Calculus - Area of a curve

I've got

(15 + 18 - 9) - (-5 + 2 + 1/3)

And the rectangle is 4 x 2 = 8.

B


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

#10 2012-04-25 08:32:54

amberzak
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Re: Calculus - Area of a curve

Could you go through step by step. I'm now completely lost.

The answer on my answer sheet, by the way, is 10 and 2/3


Don't think outside the box. Think there is no box

#11 2012-04-25 08:42:06

anonimnystefy
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Re: Calculus - Area of a curve

Hi amberzak

Do you know the formula for finding area between graphs of two functions?


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#12 2012-04-25 08:53:37

bob bundy
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Re: Calculus - Area of a curve

hi amberzak,

LATER EDIT.  THERE IS AN ERROR WITH MY INTEGRATION HERE.  SEE POST #20

I'll have a go.  I've made a diagram below and shaded some regions.

You question asks for the region I've coloured red.

When you do



you will get the area below the curve down to the x axis.  That's what area type integration does. 
(Why is a much longer post for another day I think!)

So the integration gives an answer that is too big.  Take off 8 for the green rectangle and you should get the right result.

So let's check the integration







Bob


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You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

#13 2012-04-25 08:56:46

anonimnystefy
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Re: Calculus - Area of a curve

Hi Bob

That doesn't give a precise answer. And the precise one is even easier than what you did there. You just subtract the two functions (the quadratic and the linear one) and integrate from -1 to 3.


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#14 2012-04-25 08:58:24

bob bundy
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Re: Calculus - Area of a curve

hi Stefy,

I'm glad you are looking at this too.

I'm getting an answer of 18 and 2/3.  Any ideas?

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

#15 2012-04-25 09:00:10

anonimnystefy
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Re: Calculus - Area of a curve

Well,you did say to take off 8.


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#16 2012-04-25 09:03:39

anonimnystefy
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Re: Calculus - Area of a curve

Sorry for double posting.

You integration is not correct!!!


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#17 2012-04-25 09:03:52

bob bundy
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Re: Calculus - Area of a curve

Yes, but that was my answer after taking off 8.

ie.  26 and 2/3 take 8

??

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

#18 2012-04-25 09:04:42

anonimnystefy
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Re: Calculus - Area of a curve

Look at the post right above your new one.


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#19 2012-04-25 09:12:40

amberzak
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Re: Calculus - Area of a curve

Why is the integration not right? Is it because it's missing the denominators for the 2x3^2 and 2x-1^2

(sorry if I'm not making a lot of sense. I've been doing calculus all day)


Don't think outside the box. Think there is no box

#20 2012-04-25 09:14:18

bob bundy
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Re: Calculus - Area of a curve

You integration is not correct!!!

I see it now.  I've spent all day driving and gardening (big tree to cut down) so my brain is not at its best.  Here is a correction to post #12.



 




= 18 and 2/3

less 8 = 10 and 2/3

My apologies for the error before.  I need some sleep.  sleep

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

#21 2012-04-25 09:14:27

anonimnystefy
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Re: Calculus - Area of a curve

No,he didn't integrate the term 2x correctly.


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#22 2012-04-25 09:18:00

bob bundy
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Re: Calculus - Area of a curve

Ahead of you for the first time.  see post 20

Thanks for your help Stefy.  I'm off to bed.  Bye.  wave

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

#23 2012-04-25 09:19:46

anonimnystefy
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Re: Calculus - Area of a curve

I saw it. I still do not get why the two small parts next to the rectangle don't mess it up.


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#24 2012-04-25 09:21:46

amberzak
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Re: Calculus - Area of a curve

Thanks guys. That's really helpful.

I'll be coming on tomorrow with your questions big_smile


Don't think outside the box. Think there is no box

#25 2012-04-25 09:22:48

amberzak
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Re: Calculus - Area of a curve

Anon, I was wondering that as well, actually. I might ask my teacher than tomorrow.


Don't think outside the box. Think there is no box

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