Math Is Fun Forum

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?

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

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)

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)

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)

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Area between functions

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

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.

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)

Nils-Ake

Member

Registered: 2008-04-15

Posts: 20

Re: Area between functions

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).

Daniel123

Member

Registered: 2007-05-23

Posts: 663

Re: Area between functions

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

Nils-Ake

Member

Registered: 2008-04-15

Posts: 20

Re: Area between functions

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.

Nils-Ake

Member

Registered: 2008-04-15

Posts: 20

Re: Area between functions

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 :)

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 .

No problem, sorry about the silly mistake originally.

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.

Nils-Ake

Member

Registered: 2008-04-15

Posts: 20

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 :).

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

Daniel123

Member

Registered: 2007-05-23

Posts: 663

Re: Area between functions

Yayy