Arc length & area of surface of revolution

[hiy312]

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 θ - cos 2θ, y = 2 sin θ - sin 2θ " 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!

 

[DryRun]

Hi hiy312

You should draw the graph to understand the region that makes up the surface when rotated about the x-axis. See the attached picture.

Now, you just find the polar equation of the cardioid. Here is what i get: $r=\sqrt{5-4\cos 3\theta}$ and convert to Cartesian form.

Then, you can find the area of the upper half of the solid by projecting onto the xy-plane, and multiply by 2, to get the total area.

Edit: I just realized that with the volume, you'll have to consider a z variable into the equation of the surface of revolution.

 

[HallsofIvy]

I'm glad to know that you have tried (many times!). Please show what you have doine so we won't repeat it.

 

[DryRun]

I'm glad to know that you have tried (many times!). Please show what you have doine so we won't repeat it.

Agreed.