1. ## Integral problem

How do you find the surface area of the part of the plane $3x+y+z=4$ that lies inside the cylinder $x^2+y^2=4$

2. Originally Posted by qwesl
How do you find the surface area of the part of the plane $3x+y+z=4$ that lies inside the cylinder $x^2+y^2=4$
Let $f(x,y) = z = 4 - 3x - y$

Then, $f_x = -3$ and $f_y = -1$. The cylinder intersects the $xy$-plane at the circle of radius 2 centered at the origin, that is, over the region where $- \sqrt{4 - x^2} \le y \le \sqrt{4 - x^2}$ and $-2 \le x \le 2$.

The surface area is given by $A = \int_{-2}^2 \int_{- \sqrt{4 - x^2}}^{\sqrt {4 - x^2}} \sqrt{(f_x)^2 + (f_y)^2 + 1}~dy~dx = \int_{-2}^2 \int_{- \sqrt{4 - x^2}}^{\sqrt {4 - x^2}} \sqrt{11}~dy~dx$ $= \int_0^{2 \pi} \int_0^2 \sqrt{11}r~dr ~d \theta$

I leave the rest to you