Question is related to line integration, work done by force from book Erwin krezyick advance maths

Question

I cannot understand how to write my function in the form of r(t)

Can u please solve these questions and teach me?

Q:calculate intgeral of F(r(t)). r'(t)

 

 

Qno1. F=[xy,x²y²] , C the straight line segment from (0,2) to (2,0)

 

Qno2.F=[(x-y)², (y-x)²], C; xy=1  1≤x≤4

 

Answer 1

For Question 1

Use the following parameterization:

 

r(t) = <x₀,y₀>(1-t) + <x₁,y₁>t

 

Where (x₀,y₀) be the starting point of the line segment.
And (x₁,y₁) be the end point of the line segment.

 

(Keep in mind <x₀,y₀> is a vector = x₀ i + y₀ j)

 

In this case, (x₀,y₀) = (0,2) and (x₁,y₁) = (2,0)

 

r(t)

= <0,2>(1-t) + (2,0)t

= <0, 2(1-t)> + <2t, 0>

= <2t, 2-2t>

 

dr/dt = <2, -2> dt