Homework Help: Finding integral of x2 and y2

1. Jun 17, 2012

Mathpower

1. The problem statement, all variables and given/known data
I would like to know how to integrate a function that has y^2 in it.

2. Relevant equations
eg: (x^2)/5 + (y^2) +900 = 0

3. The attempt at a solution
y = ((x^2)/5 - 900)^(1/2)
Area = ∫((x^2)/5 - 900)^(1/2)
Chain rule in reverse somehow?
= (((x^2)/5 - 900)^(3/2))/(1.5)
= No idea!

Last edited: Jun 17, 2012
2. Jun 17, 2012

oli4

Hi Mathpower,
there must be something wrong in your relevant equations
the attempted solution is wrong respect to it, but just looking at your initial equation, you have a sum of squares + 900 = 0, that can't be solved since the left side is >=900

Cheers

3. Jun 17, 2012

Mathpower

OK thank you for your help! What I am attempting to do is to learn how to do implicit differentiation in reverse.
Let me change the question:

1. The problem statement, all variables and given/known data
I would like to know how to integrate a function that has y^2 in it.

2. Relevant equations
eg: (x^2)/5 + (y^2) -1 = 0

3. The attempt at a solution
y = ((x^2)/5 - 1)^(1/2)
Area = ∫((x^2)/5 - 1)^(1/2)
Chain rule in reverse somehow?
= (((x^2)/5 - 1)^(3/2))/(1.5)
= No idea

4. Jun 17, 2012

algebrat

it is hard to figure out what you wanted to do. did you want to calculate the area? where? in the first quadrant?

Try a change of variables [STRIKE]u^2=x^2/5[/STRIKE] (see comment below), then you'll have a nice circle. then you could do a change of variable to polar coordinates.

If you don't want to do that, then you can use trig subsitution when you have roots and squared terms.

Last edited: Jun 17, 2012
5. Jun 17, 2012

Infinitum

That should be,

$$y = \pm \sqrt{1-\frac{x^2}{5} }$$

So you would basically have two integrals to do,

$$\int y dy= \int \sqrt{1-\frac{x^2}{5} } dx$$

And,

$$\int y dy= \int -\sqrt{1-\frac{x^2}{5} } dx$$

Which one you have to integrate, and with what limits, depends on what you wish to find out, as algebrat observed.

6. Jun 17, 2012

algebrat

EGADS!!! I should have said u=x/√5 me thinks!

7. Jun 17, 2012

Mathpower

Oops I kind of mixed up what I wanted to learn with the actual test question that I had. My apologies!
If possible please show me how to do the indefinite integral (not the definite integral as my working had previously suggested) of (x^2)/5 + (y^2) -1 = 0

I am sure that I can pick up how to find the area after that. Oh...and let me know how I can post a message like Infinitum (ie: the math writing style...if you know what I mean)

8. Jun 17, 2012

Infinitum

For area, integrate each of those expressions above, separately. Use the substitution, $u= \frac{x}{\sqrt{5}}$ You will have two answers, opposite in sign.

For only the indefinite integral, you can integrate the expression,

$$\int y^2 dy = \int (1-\frac{x^2}{5})dx$$

But this, you cannot use to find area.

As for the math, its called LaTeX. You can learn about it here : https://www.physicsforums.com/showthread.php?t=546968

Last edited: Jun 17, 2012
9. Jun 17, 2012

Mathpower

WOW! Thank you so much for your help! Amazing that I am getting help from someone in USA!
oh...so:
Let: $$u^2 = (x^2 /5)$$
$$\sqrt{1-u^2}$$
Indefinite integral:
$$\int\sqrt{1x-u^2}$$
=$$2 (1x-u^2)^(-0.5) . (2u)$$
=$$4u/\sqrt(1x-u^2)$$
=$$(4x^2/5)/(1x-(x^2/5))^0.5$$
=4x^2/(5^(0.5)(5x-x^2))

I get the feeling I haven't done it correctly! The problem is I haven't been taught the whole integration topic yet...but I do know differentiation...and that doesn't really help in this situation!

Last edited: Jun 17, 2012
10. Jun 17, 2012

Infinitum

Your steps after $\int \sqrt{1-u^2}$ aren't clear..

When you substitute a variable, you need to change the differential coefficient accordingly. You will have,

$$u = \frac{x}{\sqrt{5}}$$

$$du = \frac{dx}{\sqrt{5}}$$

Hence,

$$dx = \sqrt{5} du$$

Which you need to use in place of dx.

Also, once you reached $$\int \sqrt{1-u^2} du$$
You can do another substitution, $t=sin(u)$ and apply a similar technique as above. Then put back the t and u, so that you get x.

11. Jun 17, 2012

HallsofIvy

Why would you say that? I had not noticed that people from the USA refused to help others.

12. Jun 17, 2012

algebrat

Haha, maybe Mathpower just meant that they are far away from the USA, and it is amazing that the internet is bringing everyone together.

(You have assumed the negation is as you said.)

Perhaps we should start another thread on whether the USA is really helping anyone.

13. Jun 18, 2012

Mathpower

Oops I made a typo... I have corrected it in the above quote. Is it correct now?

Last edited: Jun 18, 2012
14. Jun 18, 2012

Mathpower

Hold on...hold on... Talk about jumping to conclusions my friend!
I never even remotely meant neither suggested what you interpreted.
What I meant to say was that, it is quite amazing how someone 1000 's of kilometers away can help me!

15. Jun 18, 2012

Infinitum

Why the negative exponent? And how did you get the 'x' term from?? You substituted u to get rid of x... Also, you should read post #10 again

16. Jun 18, 2012

Mathpower

Ok ...woah ..what have I done!
I look back at my working now and I am shocked! I somehow managed to mix up differentiation with integration...ie: when differentiating we subtract 1 off the power, hence 0.5-1=-0.5
Lol!
But atleast I know why I did it and our class got taught integrating by substituting today! So I do know how to do it now...a very big thank you for your help.

Oh and just as a side note: I guess Americans really are so patriotic after all!

17. Jun 18, 2012

dimension10

Let u=x/sqrt(5). Then use trig substitution.

18. Jun 19, 2012

Infinitum

Good to know you got it clear now