Arc length & area of surface of revolution

  1. 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!
  2. jcsd
  3. sharks

    sharks 837
    Gold Member

    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.

    Attached Files:

    Last edited: Jun 4, 2012
  4. HallsofIvy

    HallsofIvy 40,946
    Staff Emeritus
    Science Advisor

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

    sharks 837
    Gold Member

    Agreed. :smile:
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