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Math Help - Integral problem

  1. #1
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    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
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by qwesl View Post
    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
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