# Thread: integration...should i use polar transformation?

1. ## integration...should i use polar transformation?

Evaluate $\iint\limits_R \frac{1}{(1 + x^2 + y^2)^\frac{3}{2}} dx dy$
where $R$ is the region bounded by the straight lines $y=0 , x=1 , y=x$

if i use polar transformation, what will be the ranges of integration? or is there any way other than polar transformation?

2. I think I'd stay in cartesian coordinates for this one. What is your integral, including limits?

3. writing in terms of limits, the integration becomes $\int_0^1\int_0^x \frac{1}{(1 + x^2 + y^2)^\frac{3}{2}} dy dx$

how to evaluate this?

4. Right. So the $y$ indefinite integral is of the form

$\displaystyle\int\frac{dy}{(a^{2}+y^{2})^{3/2}}.$

How would you compute that antiderivative?

5. we substitute $y = a \tan\theta$, which gives the final value of the above integral as $\frac{1}{a^2}\sin y$, ie, $\frac{sin x}{1+x^2}$. right?? then what?

6. I think you may have skipped too many steps there. I agree with the substitution, but I get a different result. How do the integrand and differential change with your trig substitution?

7. we substitute $y = a \tan\theta$, which gives the final value of the above integral as $\frac{1}{a^2} \frac{y}{\sqrt{y^2+a^2}}$. ok? that is $\frac{x}{(1+x^2)(\sqrt{1+2x^2})}$. right??

8. The final result is correct. So, now what? You've got

$\displaystyle\int_{0}^{1}\frac{x}{(1+x^2)(\sqrt{1+ 2x^2})}\,dx.$

Ideas?

9. here we substitute $x^2 = z$ then $1 + 2z = t^2$ this gives us the final result as $\arctan\sqrt{2x^2 + 1}$. am i right?

10. Not sure I'm following your steps here. The final result is incorrect, actually. It should be arctan.

11. ya...sorry..it will be tan inverse....just a little typing mistake.....sorry for that

i have made the correction in the previous post. so the answer comes to be $\arctan \sqrt{3} = \frac{\pi}{3}$

12. Don't forget the lower limit.

13. oooops....so much silly mistakes...... yeah. the answer is then $\frac{\pi}{3}-\frac{\pi}{4} = \frac{\pi}{12}$.....

many many thanks:d

14. There you go. Nice work.

One way to avoid silly mistakes is to pretend that someone is holding a gun to your temple, and if you make a mistake, he's going to pull the trigger. Or, you could work at a high-energy physics lab, with all those 10,000V wires running around. If you make a mistake, you're dead.

So there are some nice, cheerful ways to avoid mistakes.