Sunday, 3 November 2013

calculus - Find the slant area of a cone



Question: Find the slant curved area of the surface of revolution of a cone of semi-vertical angle α and base circle of radius a by revolving about the X-axis.



I tried using r=acscθ and integrating from θ=0 to θ=α, but the answer is wrong.



Help me with the correct equation and the limits.


Answer




You should be careful to look to what you are integrating.



Consider a differential triangular area in yellow color on the slant side as shown:



enter image description here



dA=12asinαadθ



Integrating




dA=2π012asinαadθ=πa2sinα.



Also see how the standard slant area formula A=πaL is derived with L=asinα.


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