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Math Help - help! vector calculus!

  1. #1
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    Exclamation help! vector calculus!

    how to do this problem?

    Use a line integral to find the area of the region
    R bounded by the graphs of y = 2x + 1 and

    y = 4 x^2
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  2. #2
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    Quote Originally Posted by greencheeseca View Post
    how to do this problem?


    Use a line integral to find the area of the region R bounded by the graphs of y = 2x + 1 and
    y = 4 x^2
    Are you sure what I red bolded is correct?
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  3. #3
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    Green's theorem says that
    \oint f(x,y)dx+ g(x,y)dy = \int\int \left(\frac{\partial g}{\partial x}- \frac{\partial f}{\partial y}\right) dx dy

    Since the area of a region is just \int\int dx dy, choosing any f and g such that \frac{\partial g}{\partial x}- \frac{\partial f}{\partial y}= 1 will convert that double integral for area into a line integral over the boundary. A simple and valid choice is g(x,y)= x, f(x,y)= 0. The area is given by \int x dy .

    You will have to convert both parabola and line to parametric equations. Be careful about integrating in the right direction- you must integrate counter-clockwise around the boundary.

    Frankly, this is much harder than just integrating the difference of the two functions!
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  4. #4
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    yea, i know it's hard cause i havent' been able to solve it ^^ can anyone show me the solution to this problem so i can see how it's done?
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  5. #5
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    Show us what you have tried. Did you write parametric equations for the line and parabola? It's straight forward just to use x itself as parameter. On the line, y= 2x+1, dy= 2dx and on the parabola, y= 4- x^2, dy= -2xdx[/tex]. At what points do the line and parabola intersect?

    Actually, looking at this more closely, I take back what I said before- the integrals are not difficult at all!
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  6. #6
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    how are u pasting these mathematical symbols here?? i'd like to post what i tried but i can't type the symbols here ^^



    Quote Originally Posted by HallsofIvy View Post
    Green's theorem says that
    \oint f(x,y)dx+ g(x,y)dy = \int\int \left(\frac{\partial g}{\partial x}- \frac{\partial f}{\partial y}\right) dx dy

    Since the area of a region is just \int\int dx dy, choosing any f and g such that \frac{\partial g}{\partial x}- \frac{\partial f}{\partial y}= 1 will convert that double integral for area into a line integral over the boundary. A simple and valid choice is g(x,y)= x, f(x,y)= 0. The area is given by \int x dy .

    You will have to convert both parabola and line to parametric equations. Be careful about integrating in the right direction- you must integrate counter-clockwise around the boundary.

    Frankly, this is much harder than just integrating the difference of the two functions!
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  7. #7
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    Using LaTex. You can click on the "sigma" button third from the right just above the window you are typing in or just type [ math][/ math] (without the spaces I had to put in so you would be able to see those) yourself and put the code between them. There is a LaTex tutorial thread at http://www.mathhelpforum.com/math-help/latex-help/ but you can also see the code used for a particular formula by clicking on the formula.
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