just making sure I understand it....
\[S=\int_{a}^{b} 2\pi y \sqrt{1+{(\frac{dy}{dx})}^2} dx\]
\[S=\int_{c}^{d} 2\pi y \sqrt{1+{(\frac{dx}{dy})}^2} dy\]
\[S=\int_{a}^{b} 2\pi x \sqrt{1+{(\frac{dy}{dx})}^2} dx\]
One is for revolution about the x axis...the other for revolution about the y- axis...and then we have one more equation....why?

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what the devil is this?

Area of a Surface of Revolution

surface area ?

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