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Multivariable Calculus

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Difficult

Second Type Line Integral on Half Ellipse

MVCALC-0YEKTE

Evaluate:

$$I=\int_C \frac{y-z}{2}dx+\frac{z-x}{2}dy+\frac{x-y}{2}dz$$

...in which: $C$ is the intersection of cylinder $x^2+y^2=1$ and plane $x+z=1$ and the orientation of $C$ follows the shortest arc from $A(1,0,0)$ through $B(0,-1,1)$ to $C(1,0,2)$.

A

$\pi$

B

$\pi+1$

C

$\sqrt{\pi}$

D

$\pi-1$

E

$\sqrt{2}\pi$