Solving a Line Integral Problem: Struggling for the Right Answer

[galaxy_twirl]

Hi everyone. I ran into a minor problem while trying to solve a problem on line integral. I suspect this question to be very straight-forward as it gave the parametric equations of the curve C. However, I am still unable to get the answer for some reason. May I have someone to point out where I went wrong? Thanks!

Question:

Ah, the image didn't get uploaded.. Please refer to the attached image. Thanks! The question in question is Q1(a).

My attempt:

Since x = t² + 1 and y = t³+t , we have x(t) and y(t). Following on, I went one to apply the formula, ∫ F⋅ dr = ∫ f(x(t), y(t))||r'(t)|| dt.

I had r(t) as (t² + 1) i + (t³+t) j. And I have r'(t) = (2t) i + (3t² + 1) j, and ||r'(t)|| = √ (9t² + 1)(t² + 1).

Since the range of t is 0 ≤ t ≤ 1, I integrated over this range and my calculator returned 3.73 as the answer, but the answer in my answer key wrote 6. May I know where I had gone wrong? Thanks! :D

<In case the picture cannot be seen, I have uploaded it here.> Thanks again!

 

[Mark44]

Hi everyone. I ran into a minor problem while trying to solve a problem on line integral. I suspect this question to be very straight-forward as it gave the parametric equations of the curve C. However, I am still unable to get the answer for some reason. May I have someone to point out where I went wrong? Thanks!

Question:

Ah, the image didn't get uploaded.. Please refer to the attached image. Thanks! The question in question is Q1(a).

My attempt:

Since x = t² + 1 and y = t³+t , we have x(t) and y(t). Following on, I went one to apply the formula, ∫ F⋅ dr = ∫ f(x(t), y(t))||r'(t)|| dt.

I had r(t) as (t² + 1) i + (t³+t) j. And I have r'(t) = (2t) i + (3t² + 1) j, and ||r'(t)|| = √ (9t² + 1)(t² + 1).

Your r'(t) looks fine, but |r'(t)| doesn't.

Since the range of t is 0 ≤ t ≤ 1, I integrated over this range and my calculator returned 3.73 as the answer, but the answer in my answer key wrote 6. May I know where I had gone wrong? Thanks! :D

<In case the picture cannot be seen, I have uploaded it here.> Thanks again!

 

[td21]

Can you show me your work on the finding the gradient of f?

 

[galaxy_twirl]

To Mark44: Oh dear. I must have been hallucinating when doing Math at nearly 3am. T_T Sorry. ||r'(t)|| is √(t^2 + (1/9))(t^2 + 1).

To td21: How do I find the gradient of f? Do I just use the formula Gradient = (y₂-y₁)/(x₂-x₁)?

 

[galaxy_twirl]

However, my calculator returned 1.24 as the answer.. :/