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Math Help - area of region

  1. #1
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    area of region

    1. Sketch the region enclosed by the curves and find its area.
     = ,  = 4,  = − + 2
    2. Find the volume of the solid that results when the region enclosed by the given curves is
    revolved about the x-axis.
     =
    cos,  =
    
    4
    ,  =
    
    2
    ,  = 0
    3. Solve the initial value problem by separation of variables
    
    
    =
    4
     + cos
    , 1 = 
    4. Solve the initial value problem by the method of integrating factors
    
    
    − = , 0 = 3
    5. Find the equation of the plane that contains the points 1,−2,0, 3,1,4 and 0,−1,2.
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  2. #2
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    Quote Originally Posted by res88 View Post
    ...
    5. Find the equation of the plane that contains the points 1,−2,0, 3,1,4 and 0,−1,2.
    I'm sorry but I can't read your text

    to #5:

    I assume that you mean P(1, -2, 0), Q(3, 1, 4) and R(0, -1, 2).

    You need a fixed point (I take P) and 2 direction vectors:

    \overrightarrow{PQ}=\overrightarrow{OQ}-\overrightarrow{OP}= \langle 2, 3, 4\rangle

    \overrightarrow{PR}=\overrightarrow{OR}-\overrightarrow{OP}= \langle -1, 1, 2\rangle

    Then the equation of the plane is:

    \vec v = \langle1,-2,0  \rangle+s \cdot \langle 2, 3, 4\rangle + t \cdot \langle -1, 1, 2\rangle

    As a parametric system of equations:
    \begin{array}{l}x=1+2s-t \\ y=-2+3s+t \\ z=4s+2t\end{array}

    The normal vector of the plane is:

    \langle 2, 3, 4\rangle \times \langle -1, 1, 2\rangle=\langle 2, -8, 5\rangle

    And therefore the equation of the plane in normal form (keep in mind that the point P belongs to the plane!):

    2x-8y+5z-18=0
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