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Math Help - Help setting up two line integrals

  1. #1
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    Help setting up two line integrals

    How do I set up these line integrals?

    First problem:

    Line integral of y^3dx + x^2dy where C is the arc of the parabola x=1-y^2 from (0,-1) to (0,1).

    Second problem:

    Line integral of ydx + (x+y^2)dy where C is the ellipse 4x^2 + 9y^2 = 36, with counterclockwise rotation
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    Re: Help setting up two line integrals

    Quote Originally Posted by NickE View Post
    How do I set up these line integrals?

    First problem:

    Line integral of y^3dx + x^2dy where C is the arc of the parabola x=1-y^2 from (0,-1) to (0,1).

    Second problem:

    Line integral of ydx + (x+y^2)dy where C is the ellipse 4x^2 + 9y^2 = 36, with counterclockwise rotation
    Have you used Green's Theorem before?

    \displaystyle \begin{align*} \oint_C{\left(L\,dx + M\,dy\right)} &= \int{\int_D{\left(\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y}\right)\,dx}\,dy} \end{align*}

    This will help you with the second problem because it's a closed contour.
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    Re: Help setting up two line integrals

    Thanks this helps a lot. I knew there was a way to convert a line integral in that form to a double integral but could not find it in my notes.

    Edit:
    I still need help. I've found my notes on green's theorem but there are no examples on how to apply it to ellipse. I tried turning problem 2 it into a double integral and came up with the double integral of 0. What am I doing wrong. I know I'm probably overthinking this.
    Last edited by NickE; July 28th 2012 at 09:59 AM.
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    Re: Help setting up two line integrals

    Quote Originally Posted by NickE View Post
    How do I set up these line integrals?

    First problem:

    Line integral of y^3dx + x^2dy where C is the arc of the parabola x=1-y^2 from (0,-1) to (0,1).
    Have you not had any instruction in this? Since x= 1- y^2, dx= -2ydy so y^3dx+ x^2dy= y^3(-2ydy)+ (1- y^2)^2dy= -2y^4dy+ (1- 2y^2+ y^4)dy= (-y^4- 2y^2+ 1)dy. Since y is going from -1, to 1, that integral is \int_{-1}^1 (-y^4- 2y^2+ 1)dy.

    Second problem:

    Line integral of ydx + (x+y^2)dy where C is the ellipse 4x^2 + 9y^2 = 36, with counterclockwise rotation
    This is a little harder because neither x nor y is a function of the other so we have to set up a "parameterization". Fortunately, there is a "standard" parameterization of an ellipse like this: x= 3cos(t), y= 2sin(t). Do you see why? 4x^2+ 9y^2= 4(9cos^2(t))+ 9(4sin^2(t))= 36(cos^2(t)+ sin^2(t))= 36.

    Now, from x= 3cos(t), dx= -3 sin(t)dt and from y= 2sin(t), dy= 2cos(t)dt. Going completely around the ellpse, t goes from 0 to 2\pi.
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    Re: Help setting up two line integrals

    Thanks that helps me understand it better. I have had instruction in this but I was a more general lecture. We only worked a few examples but it wasn't enough for me to fully grasp how to work any problem. The best way for me to learn anything is to do it repetitively and to do different kinds of examples.
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    Re: Help setting up two line integrals

    If you use the parameterisation given by HallsOfIvy, with \displaystyle \begin{align*} x = 3\cos{t} \implies dx = -3\sin{t}\,dt \end{align*} and \displaystyle \begin{align*} y = 2\sin{t} \implies dy = 2\cos{t}\,dt \end{align*} with \displaystyle \begin{align*} 0 \leq t \leq 2\pi \end{align*}, then

    \displaystyle \begin{align*} \oint_C{\left[ y\,dx + \left(x + y^2\right)dy \right]} &= \int_0^{2\pi}{\left\{ 2\sin{t}\left(-3\sin{t}\,dt\right) + \left[ 3\cos{t} + \left(2\sin{t}\right)^2 \right]\left( 2\cos{t}\,dt \right) \right\} } \\ &= \int_0^{2\pi}{ -6\sin^2{t}\,dt + \left(6\cos^2{t} + 8\cos{t}\sin^2{t}\right) dt } \\ &= \int_0^{2\pi}{\left(6\cos{2t} + 8\cos{t}\sin^2{t}\right)dt } \\ &= \left[ 3\sin{2t} + \frac{8}{3}\sin^3{t} \right]_0^{2\pi} \\ &= 0 \end{align*}


    If you use Green's Theorem

    \displaystyle \begin{align*} L &= y \implies \frac{\partial L}{\partial y} = 1 \\ M &= x + y^2 \implies \frac{\partial M}{\partial x} = 1 \end{align*}

    So

    \displaystyle \begin{align*} \oint_C{ \left[ y\,dx + \left(x + y^2\right)dy \right] } &= \int{\int_D{\left(1 - 1 \right)dx}\,dy} \end{align*}

    We have the ellipse \displaystyle \begin{align*} 4x^2 + 9y^2 &= 36 \end{align*}. If we use the Change of Variables \displaystyle \begin{align*} x = \frac{u}{2} \end{align*} and \displaystyle \begin{align*} y = \frac{v}{3} \end{align*}, the Jacobian is

    \displaystyle \begin{align*} \frac{\partial \left(x, y \right)}{\partial \left(u, v \right)} &= \left| \begin{matrix} \frac{\partial x}{\partial u} & \frac{\partial x }{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v}  \end{matrix} \right| \\ &= \left| \begin{matrix} \frac{1}{2} & 0 \\ 0 & \frac{1}{3} \end{matrix} \right| \\ &= \frac{1}{6} \end{align*}

    and the ellipse transforms to the circle given by

    \displaystyle \begin{align*} 4x^2 + 9y^2 &= 36 \\ 4\left(\frac{u}{2}\right)^2 + 9\left(\frac{v}{3}\right)^2 &= 36 \\ u^2 + v^2 &= 36 \end{align*}

    and the integral gets transformed to

    \displaystyle \begin{align*} \int{\int_D{0\,dx}\,dy} &= \int{\int_S{0\cdot \frac{1}{6}\,du}\,dv} \\ &= \int{\int_S{0\,du}\,dv} \end{align*}

    And the region, now being a circle, should be transformed to polar coordinates. When this happens, we have the bounds \displaystyle \begin{align*} 0 \leq r \leq 6 \end{align*} and \displaystyle \begin{align*} 0 \leq \theta \leq 2\pi \end{align*}, which gives

    \displaystyle \begin{align*} \int_0^{2\pi}{\int_0^6{0\cdot r\,dr}\,d\theta} &= \int_0^{2\pi}{\int_0^6{0\,dr}\,d\theta} \\ &= \int_0^{2\pi}{\left[C_1\right]_0^6 d\theta} \\ &= \int_0^{2\pi}{0\,d\theta} \\ &= \left[C_2\right]_0^{2\pi} \\ &= 0 \end{align*}
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    Re: Help setting up two line integrals

    The original question was about doing the line integrals. Yes, the itegral around a closed curve can be done as a double integral but I don't think that was what was being asked.

    Oh, and when you have \int_A\int 0 dydx, no matter how complicated A is, I hardly think you need to convert to a simpler region! 0 has integral 0 over any region!
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    Re: Help setting up two line integrals

    Quote Originally Posted by HallsofIvy View Post
    The original question was about doing the line integrals. Yes, the itegral around a closed curve can be done as a double integral but I don't think that was what was being asked.

    Oh, and when you have \int_A\int 0 dydx, no matter how complicated A is, I hardly think you need to convert to a simpler region! 0 has integral 0 over any region!
    I was writing in response to replies by the poster, saying he wasn't sure about the bounds etc. And yes, I know that the integral over 0 is 0, I was showing what process needs to be followed in any case.
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