(All Steps) (Cycloid and line integrals) Let C be the cycloid in the plane given by the vector-valued function r(t)=(t- sin t) i+(1- cos t) j, 0 ≤q t

Question: (Cycloid and line integrals) Let $C$ be the cycloid in the plane given by the vector-valued function

\[\mathbf{r}(t)=(t-\sin t) \mathbf{i}+(1-\cos t) \mathbf{j}, \quad 0 \leq t \leq 2 \pi\]

Find the area of the region enclosed by $C$ and the $x$ -axis.
Let $\mathbf{F}(x, y)=\left(-y+y e^{x y}\right) \mathbf{i}+\left(x+x e^{x y}\right) \mathbf{j} .$ Evaluate $\int_{C} \mathbf{F} \cdot d \mathbf{r}$.

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(All Steps) (Cycloid and line integrals) Let C be the cycloid in the plane given by the vector-valued function r(t)=(t- sin t) i+(1- cos t) j, 0 ≤q t

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