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#1 2015-07-20 22:54:54

Agnishom
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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

zetafunc
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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
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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

Agnishom
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Re: Definite Integrals of Inverse Functions

No, I cannot. What is the numerical answer coming to?

zetafunc wrote:

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
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From: Bumpkinland
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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

zetafunc
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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

Agnishom
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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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#8 2015-07-23 05:34:12

zetafunc
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Re: Definite Integrals of Inverse Functions

Correct. You should be able to complete the problem now, with your answer of the form you wrote in your original post.

Last edited by zetafunc (2015-07-23 05:42:00)

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#9 2015-07-23 09:41:54

bobbym
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From: Bumpkinland
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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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#10 2015-07-23 10:23:26

zetafunc
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Re: Definite Integrals of Inverse Functions

What answer are you getting?

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#11 2015-07-23 13:15:01

bobbym
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From: Bumpkinland
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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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#12 2015-07-24 22:52:09

zetafunc
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Re: Definite Integrals of Inverse Functions

bobbym wrote:

Hi;

I am getting:

I agree with your answer.

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#13 2015-07-25 03:34:06

bobbym
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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

Agnishom
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From: Riemann Sphere
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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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#15 2015-07-25 07:14:28

zetafunc
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Re: Definite Integrals of Inverse Functions

There are other ways of finding areas besides integration.

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#16 2015-07-25 07:22:53

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Definite Integrals of Inverse Functions

Hi;

0KjLAM9.png

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

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
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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
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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

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
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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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#20 2015-07-25 07:51:19

zetafunc
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Re: Definite Integrals of Inverse Functions

Agnishom wrote:

If I know the coordinates of C,  yes

You know x. So what's f(x)?

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#21 2015-07-25 08:08:06

bobbym
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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}] // Together

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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#22 2015-07-26 01:01:13

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
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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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#23 2015-07-26 01:24:45

zetafunc
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Re: Definite Integrals of Inverse Functions

What are your limits of integration?

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#24 2015-07-26 06:23:23

bobbym
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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
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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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