Math Is Fun Forum

amberzak

Member

Registered: 2012-03-16

Posts: 80

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.

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.

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

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

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.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

amberzak

Member

Registered: 2012-03-16

Posts: 80

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)

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

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

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.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

amberzak

Member

Registered: 2012-03-16

Posts: 80

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

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Calculus - Area of a curve

Correct! Very good!

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

amberzak

Member

Registered: 2012-03-16

Posts: 80

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

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

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

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

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

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

amberzak

Member

Registered: 2012-03-16

Posts: 80

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

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Calculus - Area of a curve

Hi amberzak

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

---

“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

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

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

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.

---

“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

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

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Calculus - Area of a curve

Well,you did say to take off 8.

---

“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Calculus - Area of a curve

Sorry for double posting.

You integration is not correct!!!

---

“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Calculus - Area of a curve

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

ie.  26 and 2/3 take 8

??

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Calculus - Area of a curve

Look at the post right above your new one.

---

“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

amberzak

Member

Registered: 2012-03-16

Posts: 80

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

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

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. 

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Calculus - Area of a curve

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

---

“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

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. 

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

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.

---

“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

amberzak

Member

Registered: 2012-03-16

Posts: 80

Re: Calculus - Area of a curve

Thanks guys. That's really helpful.

I'll be coming on tomorrow with your questions

---

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

amberzak

Member

Registered: 2012-03-16

Posts: 80

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