# Homework Help: Evaluate the Following Line Integral Part 2

1. Nov 1, 2011

### bugatti79

1. The problem statement, all variables and given/known data

$\displaystyle \int_c zdx+xdy+ydz$ where C is given by $t^2\vec i +t^3 \vec j +t^2 \vec k$

Can this $\displaystyle \int_c zdx+xdy+ydz$ be written as $\displaystyle \int_c z\vec i+x \vec j+y \vec k$?

I believe I need to evalute the integral $\displaystyle \int_c \vec F( \vec r (t)) d\vec r(t)$....?

How do I proceed?

2. Nov 1, 2011

### I like Serena

Hi again bugatti79!

It should be:
$\int_c (z\vec i+x \vec j+y \vec k) \cdot d\vec r(t)$

What would $d\vec r(t)$ be?

3. Nov 1, 2011

### bugatti79

How did you establish this?

From what you have shown I rearranged to find $d\vec r = \vec i dx+\vec j dy+ \vec k dz$...........?

No, that wouldnt be right as dr is a function of t.....

$d\vec r = d(t^2 \vec i +t^3\vec j+ t^2\vec k)$

Last edited: Nov 1, 2011
4. Nov 1, 2011

### I like Serena

What you have is a dot product:
$zdx+xdy+ydz=(z,x,y) \cdot (dx,dy,dz)$

The vector (z,x,y) corresponds to your function $\vec F(\vec r)$

The vector (dx,dy,dz) corresponds to your $d\vec r(t)$

The vector (x,y,z) corresponds to your $\vec r(t)$

Yes, the latter.
Can you work this out as a derivative?

5. Nov 1, 2011

### bugatti79

$\displaystyle \int_c (z\vec i+x \vec j+y \vec k) \cdot d\vec r(t)=\int_c (t^2 \vec i+t^2 \vec j+t^3 \vec k) \cdot (2t \vec i + 3t^2 \vec j + 2t \vec k)dt = \int_{0}^{1} (2t^3 +3t^4+2t^4) dt$.......?

6. Nov 1, 2011

### I like Serena

Yep!
(That was easy, was it not? )

7. Nov 1, 2011

### bugatti79

Cheers! There will be more trickier ones along the way. :-) Thanks