# Finding this line integral

## Homework Statement

Evaluate this line integral ∫ F . dr , where F = (3x2 sin y)i + (x3 cos y)j between the origin (0,0) and the point (2,4):

(a) along straight line y = 2x
(b) along curve y = x2

## The Attempt at a Solution

Part (a)
dr = dx i + dy j

∫ [ (3x2 sin y) i + (x3 cos y)j ] . [dx i + dy j ]

= ∫ (3x2 sin y)dx + (x3 cos y)dy

= ∫ d(x3 sin y) from [0,0] to [2,4]

Does this mean that this line integral is independent of the path taken?

(b) If the line integral is independent of path, you should get the same answer..

Does dr = dx i + dy j still hold given that it's a curve now? do i have to use the "distance along curve" formula:

dr = √[ 1 + (dy/dx)2 ] dx

I've looked up RHB textbook it says it's fine to simply use dr = dx i + dy j ...

Dick
Science Advisor
Homework Helper

## Homework Statement

Evaluate this line integral ∫ F . dr , where F = (3x2 sin y)i + (x3 cos y)j between the origin (0,0) and the point (2,4):

(a) along straight line y = 2x
(b) along curve y = x2

## The Attempt at a Solution

Part (a)
dr = dx i + dy j

∫ [ (3x2 sin y) i + (x3 cos y)j ] . [dx i + dy j ]

= ∫ (3x2 sin y)dx + (x3 cos y)dy

= ∫ d(x3 sin y) from [0,0] to [2,4]

Does this mean that this line integral is independent of the path taken?

(b) If the line integral is independent of path, you should get the same answer..

Does dr = dx i + dy j still hold given that it's a curve now? do i have to use the "distance along curve" formula:

dr = √[ 1 + (dy/dx)2 ] dx

I've looked up RHB textbook it says it's fine to simply use dr = dx i + dy j ...

x and y are not independent. You can't treat y as a constant when integrating dx and vice versa. Take your first path, y=2x. r=i dx+j dy=i dx+j 2*dx. Eliminate y from the integration by putting y=2x everywhere.

x and y are not independent. You can't treat y as a constant when integrating dx and vice versa. Take your first path, y=2x. r=i dx+j dy=i dx+j 2*dx. Eliminate y from the integration by putting y=2x everywhere.

I know x and y are not independent, as they are bounded by y = 2x...

But my question here is whether the integral is independent of the path taken or not since it can be reduced to ∫ d(x3 sin y)...

Dick
Science Advisor
Homework Helper
I know x and y are not independent, as they are bounded by y = 2x...

But my question here is whether the integral is independent of the path taken or not since it can be reduced to ∫ d(x3 sin y)...

Yes it is.

haruspex
Science Advisor
Homework Helper
Gold Member
2020 Award
my question here is whether the integral is independent of the path taken or not since it can be reduced to ∫ d(x3 sin y)...
You can think of F as a field resulting from the scalar potential x3 sin y. Since that is single-valued, the field must be conservative.