How would one determine the area of the plane $x+y+z=a$ that is contained within the cylinder $x^2+y^2=b^2$?
2026-03-31 20:35:33.1774989333
Generalized area of a plane within a cylinder
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Hint. The intersection is an ellipse whose area is given by the following integral (see surface integral): $$\iint_{\{x^2+y^2\leq b^2\}}\sqrt{1+ f_x^2(x,y)+f_y^2(x,y)}dxdy$$ where $f(x,y)=z=a-x-y$.
Can you take it from here?