Evaluate the line integral of f(x,y) along the curve C.
-$$f ( x , y ) = y + x , C : x ^ { 2 } + y ^ { 2 } = 100 \text { in the first quadrant from } ( 10,0 ) \text { to } ( 0,10 )$$
A) 200
B) 400
C) 0
D) 100

Question

Quizplus · Accepted Answer

The line integral of $f(x, y) = y + x$ along the curve $C: x^2 + y^2 = 100$ in the first quadrant from $(10,0)$ to $(0,10)$ can be evaluated using the parameterization of the curve. Since the curve is a quarter circle in the first quadrant, we can parameterize it using $x = 10\cos(t)$ and $y = 10\sin(t)$ where $t$ ranges from $0$ to $\frac{\pi}{2}$. The line integral formula in this context is $\int_C f(x, y) ds$, where $ds$ is the arc length differential. Substituting the parameterization into $f(x, y)$ gives $f(10\cos(t), 10\sin(t)) = 10\sin(t) + 10\cos(t)$. To find $ds$, we use the fact that $ds = \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2}dt$, which, for our parameterization, simplifies to $10dt$. Thus, the integral becomes $\int_0^{\frac{\pi}{2}} (10\sin(t) + 10\cos(t)) \cdot 10dt$, which evaluates to $200$.