Line Integral of a function over a curve in 2D-space

[ksdhart]

Hi all. I'm having troubles with a problem from my Calculus IV course. The problem states:

For exercises 21-28, evaluate the multivariate line integral of the given function over the specified curve.

22) $\displaystyle f\left(x,y\right)=e^{xy}$ with C the line x = 3y for $\displaystyle y\in \left[-1,1\right]$

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.

 

[pka]

Hi all. I'm having troubles with a problem from my Calculus IV course. The problem states:
really solving is:
$\displaystyle \displaystyle \int _{-1}^1\:e^{3t^2}\cdot \sqrt{10}dt=\sqrt{10}\cdot \int _{-1}^1\:e^{3t^2}dt$

You problem is that $\displaystyle e^{x^2}$ has a very difficult anti-derivative.

Have you tried to use a double integral?

 

[ksdhart]

You problem is that $\displaystyle e^{x^2}$ has a very difficult anti-derivative.

Have you tried to use a double integral?

Uh... I guess I could evaluate the double integral of e^(xy)... but that's not going to give me the value the problem wants, is it? Or do you mean something else? The process I followed is the one the book uses in its one example. Sorry if I'm just being obtuse and just not seeing the obvious.

 

[pka]

Uh... I guess I could evaluate the double integral of e^(xy)... but that's not going to give me the value the problem wants, is it? Or do you mean something else? The process I followed is the one the book uses in its one example. Sorry if I'm just being obtuse and just not seeing the obvious.

The point is there is no elementary integral of $\displaystyle e^{t^2}$. SORRY!