Arc length & area of surface of revolution

Hi everyone! I have two questions, one about area of surface of revolution and another is about arc length...

I really fail to do this two question despite many times of trying so I hope someone can help me

1. Find the area of the surface of revolution generated by revolving the arc of the cardioid " x = 2 cos k - cos 2k, y = 2 sin k - sin 2k " about the X-axis.

2. A warehouse is 75m long and 40m wide. A cross-section of the roof is the inverted catenary y = 31 - 10 (e^0.05x + e^-0.05x). Find the number of square metres of roofing in the warehouse. Hint: Find the arc length of the catenary and multiply this by the length of the warehouse.

I would be really grateful if you can help me(Happy)

Re: Arc length & area of surface of revolution

Re: Arc length & area of surface of revolution

Re: Arc length & area of surface of revolution

Really pro! Thanks so much Reckoner!

Quote:

Originally Posted by

**Reckoner**

But I don't understand this step, why will cos (k-2k) turn into cos k?

Re: Arc length & area of surface of revolution

Quote:

Originally Posted by

**Reckoner** Our catenary is

,

I don't understand it either... why did 10(e^0.05x + e^-0.05x) turn into 20cos h 0.05x?

Once again, thanks so much!

Re: Arc length & area of surface of revolution

Quote:

Originally Posted by

**hiy312** But I don't understand this step, why will cos (k-2k) turn into cos k?

Remember that cosine is an even function, so :

Your other question I can address later when I have more time. For now, note that I used hyperbolic functions to make the integration smoother. You could leave it in exponential form though.

Re: Arc length & area of surface of revolution

Re: Arc length & area of surface of revolution

Really helped me a lot! A million thanks to you Reckoner!!