Find the area of the surface. the part of the plane 4x + 4y + z = 16 that lies inside the cylinder x2 + y2 = 9

As a first step, convert to cylindrical coordinates. The equation is a clue to set





then have and . Meanwhile, the plane equation tells you to use



So we parameterize the part of the plane within the cylinder with the vector-valued function



The surface integral giving the area of the surface is



where denotes the surface in question, and the surface element is



We have







So the area is

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