Calculus problem #2069

Calculus Level 4

Consider the vector function given by \(F(x,y,z)=z\ \boldsymbol{i}+y\ \boldsymbol{j}-x\ \boldsymbol{k}\). Let \(C\) be the curve given by \(\boldsymbol{r}(t)=t\ \boldsymbol{i} + \sin t\ \boldsymbol{j} + \cos t\ \boldsymbol{k}\), where \(0 \leq t \leq \pi\). What is the value of

\[ N = \int_C \boldsymbol{F} \cdot d \boldsymbol{r} ? \]