Hi all. I'm having troubles with a problem from my Calculus IV course. The problem states:
I began by parameterizing the line I was given. For any given y value, x will be three times that value, so I said that \(\displaystyle r\left(t\right)=<3t,t>\). From there I took the derivative and found its magnitude.
\(\displaystyle r'\left(t\right)=<3,1>\) and \(\displaystyle \left|\left|r'\left(t\right)\right|\right|=\sqrt{3^2+1^2}=\sqrt{10}\)
I then converted the given function into a function of t. x(t) = 3t and y(t) = t, so e^(xy) = e^(3t^2). That means the integral I'm really solving is:
\(\displaystyle \displaystyle \int _{-1}^1\:e^{3t^2}\cdot \sqrt{10}dt=\sqrt{10}\cdot \int _{-1}^1\:e^{3t^2}dt\)
But here's where I get stuck, because I don't know how to integrate that function. Wolfram Alpha says the answer involves the "Imaginary Error Function," but I don't know what that is, much less what its value might be. I'm thinking I messed up somewhere, because I don't think the textbook would ask me to integrate something like that. However, I can't see any step where I might have made an error. Any help would be much appreciated.
I began by parameterizing the line I was given. For any given y value, x will be three times that value, so I said that \(\displaystyle r\left(t\right)=<3t,t>\). From there I took the derivative and found its magnitude.
\(\displaystyle r'\left(t\right)=<3,1>\) and \(\displaystyle \left|\left|r'\left(t\right)\right|\right|=\sqrt{3^2+1^2}=\sqrt{10}\)
I then converted the given function into a function of t. x(t) = 3t and y(t) = t, so e^(xy) = e^(3t^2). That means the integral I'm really solving is:
\(\displaystyle \displaystyle \int _{-1}^1\:e^{3t^2}\cdot \sqrt{10}dt=\sqrt{10}\cdot \int _{-1}^1\:e^{3t^2}dt\)
But here's where I get stuck, because I don't know how to integrate that function. Wolfram Alpha says the answer involves the "Imaginary Error Function," but I don't know what that is, much less what its value might be. I'm thinking I messed up somewhere, because I don't think the textbook would ask me to integrate something like that. However, I can't see any step where I might have made an error. Any help would be much appreciated.
