Evaluate the line integral $\int _ { C } x e ^ { x ^ { 2 } } d x + y \sin y ^ { 2 } d y$ along the triangular path consisting of the line segment from $( 0,0 )$ to $( 2,3 )$ followed by the line segment from $( 2,3 )$ to $( 1,0 )$ followed by the line segment from $( 1,0 )$ to $( 0,0 )$ .
A) $0$
B)$\frac { 25 } { 9 }$
C)$\frac { 26 } { 9 }$
D) $3$
E) $- \frac { 8 } { 3 }$
F) $- \frac { 25 } { 9 }$
G)$- \frac { 26 } { 9 }$
H) $- 3$

Question

Evaluate the line integral $\int _ { C } x e ^ { x ^ { 2 } } d x + y \sin y ^ { 2 } d y$ along the triangular path consisting of the line segment from $( 0,0 )$ to $( 2,3 )$ followed by the line segment from $( 2,3 )$ to $( 1,0 )$ followed by the line segment from $( 1,0 )$ to $( 0,0 )$ .
A) $0$ 
B)$\frac { 25 } { 9 }$
C)$\frac { 26 } { 9 }$ 
D) $3$ 
E) $- \frac { 8 } { 3 }$ 
F) $- \frac { 25 } { 9 }$ 
G)$- \frac { 26 } { 9 }$ 
H) $- 3$

Quizplus · Accepted Answer

The given line integral can be simplified by recognizing that the path of integration forms a closed loop, meaning the starting point and the ending point are the same. For conservative vector fields, the line integral over a closed loop is zero. Although the vector field here is not explicitly given, the integral's setup implies a path-independent scenario because the integral's value over a closed loop (a complete triangle in this case) results in no net change in the function being integrated. This is a property of conservative fields, but without delving into the specifics of whether the field is conservative, the key insight is that the integral sums up to zero due to the closed nature of the path and the absence of any sources or sinks within the loop that would otherwise contribute to a non-zero integral.

Evaluate the Line Integral ∫Cxex2dx+ysin⁡y2dy\int _ { C } x e ^ { x ^ { 2 } } d x + y \sin y ^ { 2 } d y∫C​xex2dx+ysiny2dy

Evaluate the Line Integral $\int _ { C } x e ^ { x ^ { 2 } } d x + y \sin y ^ { 2 } d y$