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#1 2015-07-20 22:54:54
Definite Integrals of Inverse Functions
If
Find
The answer should have a closed form like
Last edited by Agnishom (2015-07-20 22:55:06)
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#2 2015-07-21 00:56:48
Re: Definite Integrals of Inverse Functions
There are two ways to do this problem: geometrically or algebraically. (The former will give you a better idea of what's going on.)
For a geometric approach, try plotting a graph of your function and looking at which parts of the picture correspond to which integrals -- in particular, the area for your inverse function.
For an algebraic approach, make a substitution like x = f(t).
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#3 2015-07-21 09:22:37
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Definite Integrals of Inverse Functions
Can you get an inverse of that integrand?
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.
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#4 2015-07-21 12:43:37
Re: Definite Integrals of Inverse Functions
No, I cannot. What is the numerical answer coming to?
There are two ways to do this problem: geometrically or algebraically. (The former will give you a better idea of what's going on.)
For a geometric approach, try plotting a graph of your function and looking at which parts of the picture correspond to which integrals -- in particular, the area for your inverse function.
For an algebraic approach, make a substitution like x = f(t).
That does not help because if I brought in t, I'd have to change the limits of integration. Can you show me how to do it?
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.
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#5 2015-07-21 16:17:57
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Definite Integrals of Inverse Functions
No, I cannot. What is the numerical answer coming to?
That is not what I meant.
many mathematical functions do not have unique inverses.
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.
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#6 2015-07-23 01:52:26
Re: Definite Integrals of Inverse Functions
It seems that you can't get anything helpful from the limits of integration (in particular, f-¹(π/2) isn't that helpful). So try the geometric approach, which seems much easier:
-Plot a graph of f(x).
-Look at the area you're trying to compute.
-Can you see how to integrate the inverse function without finding it explicitly? (How would you graph f-¹(x) given the graph of f(x)?)
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#7 2015-07-23 03:31:17
Re: Definite Integrals of Inverse Functions
I would be needing to figure out the area between the y-axis and the curve.
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.
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#9 2015-07-23 09:41:54
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Definite Integrals of Inverse Functions
I am not getting an answer like Agnishom's suggested closed form.
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.
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#11 2015-07-23 13:15:01
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Definite Integrals of Inverse Functions
Hi;
I am getting:
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.
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#13 2015-07-25 03:34:06
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Definite Integrals of Inverse Functions
Hi;
Then that is the closed form Agnishom requires.
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.
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#14 2015-07-25 05:28:42
Re: Definite Integrals of Inverse Functions
How do I go there?
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.
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#16 2015-07-25 07:22:53
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Definite Integrals of Inverse Functions
Hi;
You need the area on the left.
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.
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#17 2015-07-25 07:44:54
Re: Definite Integrals of Inverse Functions
I still do not get it
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.
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#18 2015-07-25 07:46:02
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Definite Integrals of Inverse Functions
Do you know how to get the area on the right?
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.
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#19 2015-07-25 07:48:14
Re: Definite Integrals of Inverse Functions
If I know the coordinates of C, yes
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.
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#21 2015-07-25 08:08:06
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Definite Integrals of Inverse Functions
You know the coordinates of that rectangle.
This gets the answer.
\[Pi]/2 (1 + \[Pi]/2) - Integrate[Sin[x] + x, {x, 0, \[Pi]/2}] // TogetherIn 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.
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#22 2015-07-26 01:01:13
Re: Definite Integrals of Inverse Functions
How do I get the coordinates of that rectangl ?
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.
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#24 2015-07-26 06:23:23
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Definite Integrals of Inverse Functions
B is (pi/2,0) because that is the limit of integration given.
C is (x, sin(x)+x) which is (pi/2, sin(pi/2)+pi/2)
D is then (0,sin(pi/2)+pi/2)
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.
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#25 2015-07-26 08:18:36
- anonimnystefy
- Real Member
- From: Harlan's World
- Registered: 2011-05-23
- Posts: 16,049
Re: Definite Integrals of Inverse Functions
Hi bobbym
The y-coordinate of C needs to be pi/2, though, not the x coordinate.
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