Answer to Question #242249 in Calculus for Chapzoz

Question #242249

A cone is generated when the region is bounded by the line x=y and the vertical line x=0 and x=r is rotated about the x-axis . Use the pappus theorem to show that , Surface area of the cone is given by S=√2πr²


1
Expert's answer
2021-09-27T14:30:19-0400

Using the Pappus Theorem, we have that the surface area of the cone: "S=Ld" , where "L" the curve (blue segment) length and "d" is the distance  traveled by the centroid (red point).


We can find "L" using the Pythagoras' Theorem: "L=\\sqrt{r^2+r^2}=\\sqrt{2r^2}=r\\sqrt{2}"

"d" equals to the perimeter of the circle with the radius "r\/2" : "d=2\\pi\\frac{r}{2}=\\pi r" .

The we have "S=Ld =r\\sqrt{2}\\cdot \\pi r=\\sqrt{2}\\pi r^2"



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