# Determining the area enclosed by inverse

1. Nov 2, 2013

### Saitama

1. The problem statement, all variables and given/known data
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.

2. Relevant equations

3. 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!

2. Nov 2, 2013

### pasmith

By inspection, $f(1) = 5$ and $f(-1) = -3$. Hence $g(-3) = -1$ and $g(5) = 1$. I believe that $A$ is then
$$\int_0^1 5 - f(y)\,dy$$

Last edited: Nov 2, 2013
3. Nov 2, 2013

### Saitama

How do you get this? :uhh:

4. Nov 2, 2013

### pasmith

Ignore that; it's wrong. I edited my post.

5. Nov 2, 2013

### Saitama

I meant how you get 5-f(y)?

6. Nov 2, 2013

### pasmith

Draw a graph of $y = g(x)$ (which is by definition the curve $x = f(y)$) for $-3 \leq x \leq 5$, and identify the area A. Then express A as an integral with respect to y.

7. Nov 2, 2013

### Saitama

I honestly don't see it. Can you please elaborate some more or give me a relevant link?

8. Nov 2, 2013

### I like Serena

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. Nov 2, 2013

### Saitama

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.

10. Nov 2, 2013

### I like Serena

My bad. I meant the area enclosed by f(x) between y=-3 and y=5 (and the y-axis).

11. Nov 2, 2013

### Saitama

But for that I need the inverse of f(x).

12. Nov 2, 2013

### I like Serena

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. Nov 2, 2013

### Saitama

pasmith mentioned them above. :P

For f(x)=-3, x=-1 and for f(x)=5,x=1.

14. Nov 2, 2013

### I like Serena

So....

15. Nov 2, 2013

### Saitama

Do I have to integrate f(x) from x=-1 to 1?

16. Nov 2, 2013

### I like Serena

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. Nov 2, 2013

### Saitama

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. Nov 2, 2013

### I like Serena

Perhaps you can calculate the integral of f(x) between x=-1 and x=1?

Which area will you have calculated?

19. Nov 2, 2013

### I like Serena

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. Nov 2, 2013

### Saitama

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.