Line integral?

Question

Evaluate the line integral by the following method. 

xy dx + x^2y^3 dy 
C is counterclockwise around the triangle with vertices (0, 0), (1, 0), and (1, 4) 
(a) directly

Answer 1

Line integral by Direct Method :

Given curve C is

The triangle with vertices (0 , 0), (1, 0), and (1, 4).

C consists of three line segments ( and parameterized )

 

1) Segment C₁ is from (0,0) to (1,0) ⇒ Line equation is y = 0 and x range is in [ 0 , 1 ].

Substitute : y = 0

 

2) Segment C₂ is from (1,0) to (1,4) ⇒ Line equation is x = 1 and y range is in [ 0 , 4 ].

Substitute : x = 1

 

 

3) Segment C₃ is from (1,4) to (0,0) ⇒ Line equation is y = 4x and x range is in [ 1 , 0 ].

y = 4x  ⇒  dy = 4dx

Substitute : y = 4x and dy = 4dx

 

Going around the triangle counterclockwise we will get

C = 0 + 64 - 44

C = 20

Solution : C = 20