Determining the area enclosed by inverse

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Homework Statement


Area enclosed by y=g(x), x=-3, x=5 and x-axis where g(x) is inverse function of ##f(x)=x^3+3x+1## is A, then find [A] where [.] denotes the greatest integer function.


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The Attempt at a Solution


Honestly, I see no way to proceed here. Finding out the inverse is not possible here so I guess there is some trick to the problem.

Any help is appreciated. Thanks!
 

Answers and Replies

  • #2
pasmith
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Homework Statement


Area enclosed by y=g(x), x=-3, x=5 and x-axis where g(x) is inverse function of ##f(x)=x^3+3x+1## is A, then find [A] where [.] denotes the greatest integer function.


Homework Equations





The Attempt at a Solution


Honestly, I see no way to proceed here. Finding out the inverse is not possible here so I guess there is some trick to the problem.

Any help is appreciated. Thanks!
By inspection, [itex]f(1) = 5[/itex] and [itex]f(-1) = -3[/itex]. Hence [itex]g(-3) = -1[/itex] and [itex]g(5) = 1[/itex]. I believe that [itex]A[/itex] is then
[tex]
\int_0^1 5 - f(y)\,dy
[/tex]
 
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  • #3
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[tex]A = \int_{-3}^5 g(x)\,dx = \int_{g(-3)}^{g(5)} 5 - f(y)\,dy[/tex]
How do you get this? :uhh:
 
  • #4
pasmith
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How do you get this? :uhh:
Ignore that; it's wrong. I edited my post.
 
  • #5
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Ignore that; it's wrong. I edited my post.
I meant how you get 5-f(y)?
 
  • #6
pasmith
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I meant how you get 5-f(y)?
Draw a graph of [itex]y = g(x)[/itex] (which is by definition the curve [itex]x = f(y)[/itex]) for [itex]-3 \leq x \leq 5[/itex], and identify the area A. Then express A as an integral with respect to y.
 
  • #7
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Draw a graph of [itex]y = g(x)[/itex] (which is by definition the curve [itex]x = f(y)[/itex]) for [itex]-3 \leq x \leq 5[/itex], and identify the area A. Then express A as an integral with respect to y.
I honestly don't see it. Can you please elaborate some more or give me a relevant link?
 
  • #8
I like Serena
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Hi Pranav!

The area enclosed by ##f^{-1}(x)## between ##x=-3## and ##x=5##, is the same area as the one enclosed by ##f(y)## between ##y=-3## and ##y=5##.
It's what you get if you take the mirror image in the line y=x.

Can you find that area?
 
  • #9
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Hi ILS! :)

Hi Pranav!

The area enclosed by ##f^{-1}(x)## between ##x=-3## and ##x=5##, is the same area as the one enclosed by ##f(y)## between ##y=-3## and ##y=5##.
It's what you get if you take the mirror image in the line y=x.

Can you find that area?
##f(y)=y^3+3y+1##.

The area enclosed by f(y) between y=-3 and y=5 is ##\displaystyle \int_{-3}^5 (y^3+3y+1)dy=168##.

But this doesn't look like the right answer.
 
  • #10
I like Serena
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Hi ILS! :)

##f(y)=y^3+3y+1##.

The area enclosed by f(y) between y=-3 and y=5 is ##\displaystyle \int_{-3}^5 (y^3+3y+1)dy=168##.

But this doesn't look like the right answer.
My bad. I meant the area enclosed by f(x) between y=-3 and y=5 (and the y-axis).
 
  • #11
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My bad. I meant the area enclosed by f(x) between y=-3 and y=5 (and the y-axis).
But for that I need the inverse of f(x). :confused:
 
  • #12
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But for that I need the inverse of f(x). :confused:
Not really. You only need the boundaries for x that correspond to y=-3 and y=5.
So you need to solve f(x)=-3 and f(x)=5.
Can you?
 
  • #13
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Not really. You only need the boundaries for x that correspond to y=-3 and y=5.
So you need to solve f(x)=-3 and f(x)=5.
Can you?
pasmith mentioned them above. :P

For f(x)=-3, x=-1 and for f(x)=5,x=1.
 
  • #14
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pasmith mentioned them above. :P

For f(x)=-3, x=-1 and for f(x)=5,x=1.
So....
 
  • #16
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Do I have to integrate f(x) from x=-1 to 1?
Not exactly.
Take a look at this Wolfram picture.

You need the area enclosed between the lines x=0, y=-3, and the graph.
And that combined with the area enclosed between the lines x=0, y=5, and the graph.
 
  • #17
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Not exactly.
Take a look at this Wolfram picture.

You need the area enclosed between the lines x=0, y=-3, and the graph.
And that combined with the area enclosed between the lines x=0, y=5, and the graph.
I am not sure if I get it but I guess I will have to break the integral. I just saw the plot of ##x^3+3x+1## and it looks like there is a root between -1 and 0. Unfortunately, the root is fractional so I don't know would I have noticed that during the examination. I think I am missing something else too.
 
  • #18
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I am not sure if I get it but I guess I will have to break the integral. I just saw the plot of ##x^3+3x+1## and it looks like there is a root between -1 and 0. Unfortunately, the root is fractional so I don't know would I have noticed that during the examination. I think I am missing something else too.
Perhaps you can calculate the integral of f(x) between x=-1 and x=1?

Which area will you have calculated?
 
  • #19
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For reference, here is the Wolfram picture showing what you have to calculate according to your problem description.

Can you indicate which areas are the relevant areas?
 
  • #20
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Perhaps you can calculate the integral of f(x) between x=-1 and x=1?
I will break this integral from x=-1 to a and from x=a to 1, where a is the root. The value is negative for -1 to a is negative. Changing the sign and adding it to value obtained between a to 1 gives the area between f(x) and the x-axis. But that's not what I need.

To calculate what is required, I need the value of a but as I said before, it is fractional and would be impossible to find during the exam.
 
  • #21
LCKurtz
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Do I have to integrate f(x) from x=-1 to 1?
No. You can do it for x = -1 to 1 if you have the correct integrand(s). Have you drawn a picture of the required area? Show us your integral.

[Edit] Didn't see the second page before I posted this...
 
  • #22
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I will break this integral from x=-1 to a and from x=a to 1, where a is the root. The value is negative for -1 to a is negative. Changing the sign and adding it to value obtained between a to 1 gives the area between f(x) and the x-axis. But that's not what I need.

To calculate what is required, I need the value of a but as I said before, it is fractional and would be impossible to find during the exam.
You do not need that "a".
Take another look at the graph: http://m.wolframalpha.com/input/?i=plot[y=x^3+3x+1,x=-1,x=0,x=1,y=-3,y=5,+{x,-1,1}]&x=-1505&y=-71

What you name "a", is the y-value for x=0.
You don't need to integrate to and from "a", but to and from x=0.
 
  • #23
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You do not need that "a".
Take another look at the graph: http://m.wolframalpha.com/input/?i=plot[y=x^3+3x+1,x=-1,x=0,x=1,y=-3,y=5,+{x,-1,1}]&x=-1505&y=-71

What you name "a", is the y-value for x=0.
You don't need to integrate to and from "a", but to and from x=0.
I have shown the required area in the attachment, the shaded area is what we require, right?

I am not sure but is the graph symmetrical? It looks so.

I calculated the area on the right side of x=0. It can be found from the integral:
$$5-\int_{0}^1 (x^3+3x+1)dx=5-\frac{11}{4}=\frac{9}{4}$$

Twice of 9/4 is 9/2, hence area is 9/2=4.5, or [A]=4.

This is the answer shown by the key. Is this correct?
 

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  • #24
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I have shown the required area in the attachment, the shaded area is what we require, right?

I am not sure but is the graph symmetrical? It looks so.

I calculated the area on the right side of x=0. It can be found from the integral:
$$5-\int_{0}^1 (x^3+3x+1)dx=5-\frac{11}{4}=\frac{9}{4}$$

Twice of 9/4 is 9/2, hence area is 9/2=4.5, or [A]=4.

This is the answer shown by the key. Is this correct?
Good point.
You can calculate the area to the right of the y-axis, which is what you did.
The graph is not symmetrical, but it's close enough for our purposes.
You are only asked to give an integer approximation, which is what you did.

Good! :wink:
 
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  • #25
LCKurtz
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Good point.
You can calculate the area to the right of the y-axis, which is what you did.
The graph is not symmetrical, but it's close enough for our purposes.
You are only asked to give an integer approximation, which is what you did.

Good! :wink:
Well, I would disagree with that. If you don't prove it is symmetric, don't use it. It would be easy enough to calculate the areas of both regions with dx integrals and be sure.
 

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