Evaluate the line integral of the function over the given curve: $f ( x , y ) = x e ^ { y }$ and the curve C is given by $\vec { r } ( t ) = \langle \cos ( t ) , \sin ( t ) \rangle , 0 \leq t \leq \frac { \pi } { 2 }$
A) $e$
B) $e - 1$
C) $- 1$
D) $e ^ { \frac { \pi } { 2 } } - 1$

Question

Quizplus · Accepted Answer

To evaluate the line integral of $f(x, y) = xe^y$ over the curve $C$ given by $\vec{r}(t) = \langle \cos(t), \sin(t) \rangle$ for $0 \leq t \leq \frac{\pi}{2}$, we substitute $x = \cos(t)$ and $y = \sin(t)$ into $f$, and then integrate with respect to $t$ from $0$ to $\frac{\pi}{2}$. The derivative of $\vec{r}(t)$ with respect to $t$ is $\vec{r}'(t) = \langle -\sin(t), \cos(t) \rangle$, and the magnitude of $\vec{r}'(t)$ is $\sqrt{(-\sin(t))^2 + (\cos(t))^2} = 1$. Thus, the line integral becomes $\int_{0}^{\frac{\pi}{2}} \cos(t)e^{\sin(t)} dt$, which evaluates to $e - 1$.

Evaluate the Line Integral of the Function Over the Given f(x,y)=xeyf ( x , y ) = x e ^ { y }f(x,y)=xey

Evaluate the Line Integral of the Function Over the Given $f ( x , y ) = x e ^ { y }$