Math Is Fun Forum

  Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °

You are not logged in.

#1 2008-04-19 08:20:46

Nils-Ake
Member
Registered: 2008-04-15
Posts: 20

Area between functions

Hi! I just had a calculus exam where I was asked to calculate the area between:

I'm so disappointed cause I wasn't able to solve it :( So how would you do it?

Offline

#2 2008-04-19 08:50:16

Daniel123
Member
Registered: 2007-05-23
Posts: 663

Re: Area between functions

NOTE:  I HAVE ABSOLUTELY NO IDEA IF THE FOLLOWING IS CORRECT

Equating the two functions shows that they only meet at x=0, so your limits of integration are 3 and ∞.

Using partial fractions (which can be done by inspection if you're keen)

British school calculus tends not to cover this stuff, so I will have to use my very sketchy knowledge of limits. Apologies if this is miserably incorrect. The only method I really know of is L'Hôpital's rule, so lets go ahead and use it smile

EDIT: If you didn't know L'Hôpital's rule, you could divide the fraction out to give

, which clearly tends to 1 as x tends to infinity.

We know that the logarithm of 1 is 0, and so that disappears nicely. We are therefore left with:

Can anyone verify this for me?

Last edited by Daniel123 (2008-04-20 00:00:26)

Offline

#3 2008-04-19 13:30:58

Dragonshade
Member
Registered: 2008-01-16
Posts: 147

Re: Area between functions

I think he means the area between x=0 and x= 3 , area enclosed by the three?
Wow, its not define lol

Last edited by Dragonshade (2008-04-19 13:47:58)

Offline

#4 2008-04-19 22:22:42

Daniel123
Member
Registered: 2007-05-23
Posts: 663

Re: Area between functions

I think he means the area between x=0 and x= 3 , area enclosed by the three?

No, I don't think so. On his diagram he has shaded the region that I used yellow.

Last edited by Daniel123 (2008-04-19 23:08:18)

Offline

#5 2008-04-19 23:59:58

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

Re: Area between functions

Haha, I didn't even realise that was a link. smile

Just had a look at your working and it looks like on the fourth line, the numerator should be 4x instead of 4.

Following that through gives the end result as

.
Unfortunately that isn't finite, so I might have gone wrong somewhere too...

(Fun fact: Despite knowing and using L'Hopital for several years now, I got taught it for the first time a few days ago. The lecturer said he was thinking about cutting it from the course if we ran low on time as well.)


Why did the vector cross the road?
It wanted to be normal.

Offline

#6 2008-04-20 00:05:31

Daniel123
Member
Registered: 2007-05-23
Posts: 663

Re: Area between functions

math. How irritating. Sorry everyone.

I have just worked through the correct working, and I get the same thing as you mathsy.

Last edited by Daniel123 (2008-04-20 00:10:47)

Offline

#7 2008-04-20 00:25:24

Nils-Ake
Member
Registered: 2008-04-15
Posts: 20

Re: Area between functions

Daniel123 wrote:

No, I don't think so. On his diagram he has shaded the region that I used yellow.

Exactly what Daniel said (from x=3 to +infinity). Sorry if I wasn't clear.

My approach was to calculate the area under both functions so that:

This is what I did:

The integral diverges so the area is undefined.

I guess this would work if I was to calculate the area between f(x) and y=0, but not in this case :\ I don't know how to tell if f(x) converges on g(x).

Offline

#8 2008-04-20 00:44:57

Daniel123
Member
Registered: 2007-05-23
Posts: 663

Re: Area between functions

Are you sure that what you have done isn't correct?

Offline

#9 2008-04-20 01:19:57

Nils-Ake
Member
Registered: 2008-04-15
Posts: 20

Re: Area between functions

Daniel123 wrote:

Are you sure that what you have done isn't correct?

No, I won't be sure until April 29. But while I was working on this exercise I realized there was a teacher next to me staring at my exam. His face was like "what the hell is this guy doing?". I said to him, "this must wrong..." and then he smiled and walked away. So I assumed my answer was not correct.

Offline

#10 2008-04-20 01:41:09

Nils-Ake
Member
Registered: 2008-04-15
Posts: 20

Re: Area between functions

Daniel123 wrote:

I have just worked through the correct working, and I get the same thing as you mathsy.

Hm, so you get to the same point. Maybe I wasn't wrong in the end? Thanks a lot for your help :)

Offline

#11 2008-04-20 01:46:33

Daniel123
Member
Registered: 2007-05-23
Posts: 663

Re: Area between functions

I can't see any flaw in my (now corrected) working, and I trust mathsy too, so hopefully you were right smile.

No problem, sorry about the silly mistake originally.

Offline

#12 2008-04-21 00:25:15

moxiuming
Member
Registered: 2008-04-20
Posts: 7

Re: Area between functions

to calcute area between some functions. please follow these steps:
step1: give their graphs togher.
step2: describe the area,remember we have two type of describtions called x-type and y-type.
step3: transffer the area to integration.
step4: calcute the integration.

Offline

#13 2008-04-21 07:08:13

Nils-Ake
Member
Registered: 2008-04-15
Posts: 20

Re: Area between functions

Daniel123 wrote:

I can't see any flaw in my (now corrected) working, and I trust mathsy too, so hopefully you were right :).

The solution has just been published. We got it right! :)) thank you all for your help.

Offline

#14 2008-04-21 09:00:22

Daniel123
Member
Registered: 2007-05-23
Posts: 663

Re: Area between functions

Yayy smile

Offline

Board footer

Powered by FluxBB