Volume of solid formed by revolution of one loop of Lemniscate of bernoulli

[raghavhv]

Hello ppl. I have a problem in finding out the volume of solid formed by the revolution of one loop of lemniscate of bernoulli ( r²=a²cos2θ) about the initial line θ =0
Using the relevant forumula for the volume of the solid generated by the revolution of one loop of the polar curve about the initial line,

http://img357.imageshack.us/img357/4228/93696380jt5.jpg ,

where V is the volume of the solid and π/4 and 0 are the upper and lower limits respectively.
In this problem ,

http://img88.imageshack.us/img88/6624/19344682ul0.jpg

But i am not able to integrate further. I am kinda stuck here. What substitution should i take?Please help me.

 

[gabbagabbahey]

Hello ppl. I have a problem in finding out the volume of solid formed by the revolution of one loop of lemniscate of bernoulli ( r²=a²cos2θ) about the initial line θ =0



Using the relevant forumula for the volume of the solid generated by the revolution of one loop of the polar curve about the initial line,

http://img357.imageshack.us/img357/4228/93696380jt5.jpg ,

where V is the volume of the solid and π/4 and 0 are the upper and lower limits respectively.



In this problem ,

http://img88.imageshack.us/img88/6624/19344682ul0.jpg

But i am not able to integrate further. I am kinda stuck here. What substitution should i take?Please help me.

Try using the following trig identity:

cos(2 \theta )=2cos^2( \theta )-1

And then use a substitution like u \equiv \sqrt{2} cos( \theta )