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Math Help - please help with using double integrals to solve for mass and center of mass

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    please help with using double integrals to solve for mass and center of mass

    Find the mass and center of mass of the lamina for density = k
    where R = a triangle with vertices (0,0), (b/2,h), (b,0)

    (use double integrals to solve for both the mass and center of mass!)

    ive tried this problem a million times and i cant get the right answer :-( (and yeah i checked calcchat but i need a more detailed explanation....walk me through it please? :-) )
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    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by Skittles8768 View Post
    Find the mass and center of mass of the lamina for density = k
    where R = a triangle with vertices (0,0), (b/2,h), (b,0)

    (use double integrals to solve for both the mass and center of mass!)

    ive tried this problem a million times and i cant get the right answer :-( (and yeah i checked calcchat but i need a more detailed explanation....walk me through it please? :-) )
    First off, if \delta\!\left(x,y\right) is the density of the lamina, then M=\iint\limits_{R}\delta\!\left(x,y\right)\,dA.

    In our case, R is a triangle with vertices (0,0), (b,0) and (b/2,h). From this, we can come up with the equation of the line that connects the points (b,0) and (b/2,h) and the points (0,0) and (b/2,h). I leave it for you to verify that the equations of the lines are y=-\frac{2h}{b}x+2h and y=\frac{2h}{b}x respectively.

    Thus, it follows that if \delta\!\left(x,y\right)=k, then

    M=\iint\limits_{R}k\,dA=k\cdot\text{Area of $R$}=k\cdot\left(\tfrac{1}{2}bh\right)=\tfrac12 kbh

    Now, to find center of mass, we need to evaluate M_x=\iint\limits_{R}x\delta\!\left(x,y\right)\,dA and M_y=\iint\limits_{R}y\delta\!\left(x,y\right)\,dA.

    From there, it follows that

    \bar{x}=\frac{M_x}{M}=\frac{\displaystyle\iint\lim  its_{R}x\delta\!\left(x,y\right)\,dA}{\displaystyl  e\iint\limits_{R}\delta\!\left(x,y\right)\,dA} and \bar{y}=\frac{M_y}{M}=\frac{\displaystyle\iint\lim  its_{R}y\delta\!\left(x,y\right)\,dA}{\displaystyl  e\iint\limits_{R}\delta\!\left(x,y\right)\,dA}

    Note that M_x=k\int_0^h\int_{\frac{b}{2h}y}^{-\frac{b}{2h}y+b}x\,dx\,dy and M_y=k\int_0^h\int_{\frac{b}{2h}y}^{-\frac{b}{2h}y+b}y\,dx\,dy.

    I leave it for you to simplify the two integrals and figure out the coordinates for the center of mass.

    Can you take it from here?
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