Integrating P along Curve c: A Math Challenge

[joe:)]

Homework Statement

So I'm trying to find the integral of P = 2xyz^2 along the curve c which is defined by:

x=t^2
y=2t
z=t^3

t goes from 0 to 1

So the q says that it is the integral of P dr along c

Homework Equations

The Attempt at a Solution

So I know that this should equal the integral of P times mod r'(t) dt..

But if r = (t^2, 2t, t^3) then mod of r'(t) is pretty ugly..and I can't solve the integral..

Any pointers?

 

[joe:)]

Line Integral of Scalar Function

1. Homework Statement

So I'm trying to find the integral of P = 2xyz^2 along the curve c which is defined by:

x=t^2
y=2t
z=t^3

t goes from 0 to 1

So the q says that it is the integral of P dr along c


2. Homework Equations



3. The Attempt at a Solution

So I know that this should equal the integral of P times mod r'(t) dt..

But if r = (t^2, 2t, t^3) then mod of r'(t) is pretty ugly..and I can't solve the integral..

Any pointers?

 

[ƒ(x)]

Write P in terms of t?

 

[ideasrule]

So I know that this should equal the integral of P times mod r'(t) dt..

But if r = (t^2, 2t, t^3) then mod of r'(t) is pretty ugly..and I can't solve the integral..

Any pointers?

I have to admit I don't know either, but I'm fairly confident the teacher doesn't expect you to solve the integral using the "standard" method. Here's how complicated the improper integral is:

http://integrals.wolfram.com/index.jsp?expr=x^9*sqrt%289*x^4%2B4x^2%2B4%29&random=false

 

[gabbagabbahey]

So I know that this should equal the integral of P times mod r'(t) dt..

But if r = (t^2, 2t, t^3) then mod of r'(t) is pretty ugly..and I can't solve the integral..

Any pointers?

You might want to start by completing the square on ||\textbf{r}'(t)||^2[/tex] and then make an appropriate substitution. Even then, it looks like you will need to use integration by parts several times.

 

[berkeman]

(Two threads merged and moved to Calculus & Beyond)

Please do not multiple post questions here, Joe. Thanks.

 

[joe:)]

I have to admit I don't know either, but I'm fairly confident the teacher doesn't expect you to solve the integral using the "standard" method. Here's how complicated the improper integral is:

http://integrals.wolfram.com/index.jsp?expr=x^9*sqrt%289*x^4%2B4x^2%2B4%29&random=false

Hmm yes, complicated indeed!

Is there a theorem I can use? Greens? Stokes?

 

[gabbagabbahey]

Hmm yes, complicated indeed!

Is there a theorem I can use? Greens? Stokes?

I don't see any tricks to make this one easier. Your curve isn't closed, so it doesn't bound a surface and certainly doesn't enclose a volume so Green's and Stokes' theorems won't help. The fundamental theorem for gradients also doesn't seem to help here (I can't think of a scalar function for which \mathbf{\nabla}\Phi\cdot d\textbf{r}=Pdr...can you?) I also don't see any tricks involving the limits of integration, so it seems to me like you will have to do it the "standard way" if you are required to calculate the integral analytically.

For the "standard way", I stand by my earlier suggestion.