Evaluate the line integral of f(x,y) along the curve C.
-$$f ( x , y ) = \cos x + \sin y , C : y = x , 0 \leq x \leq \frac { \pi } { 2 }$$
A) $2 \sqrt { 2 }$
B) 2
C) 0
D) $\sqrt { 2 }$

Question

Quizplus · Accepted Answer

The line integral of $f(x, y) = \cos x + \sin y$ along the curve $C: y = x, 0 \leq x \leq \frac{\pi}{2}$ can be computed by substituting $y = x$ into the function, giving $f(x, x) = \cos x + \sin x$. The integral becomes $\int_{0}^{\frac{\pi}{2}} (\cos x + \sin x) dx$. This evaluates to $[\sin x - \cos x]_{0}^{\frac{\pi}{2}} = (1 - 0) - (0 + 1) = 2$. However, since the curve is diagonal, we need to account for the path length, which introduces a factor of $\sqrt{2}$ due to the Pythagorean theorem. Thus, the correct answer is $2\sqrt{2}$.

Evaluate the Line Integral of F(x,y) Along the Curve C f(x,y)=cos⁡x+sin⁡y,C:y=x,0≤x≤π2f ( x , y ) = \cos x + \sin y , C : y = x , 0 \leq x \leq \frac { \pi } { 2 }f(x,y)=cosx+siny,C:y=x,0≤x≤2π​

Evaluate the Line Integral of F(x,y) Along the Curve C $f ( x , y ) = \cos x + \sin y , C : y = x , 0 \leq x \leq \frac { \pi } { 2 }$