Verify Green's Theorem in a plane for
$$\ \int_C (\sin x - y^2)dx +(x-y^2)dy $$ where $\ C $ is the boundary curve of the region $R=\{(x,y)| y \ge x^2 + 1, y \leq2\}$.
Here $P=\sin x - y^2$ and $Q=x-y^2$.
$$\ \int \int_R (Q_x - P_y)dx dy = \int_{-1}^1 \int_{x^2+1}^2 (1+2y)dxdy = ...=\frac{28}{5}. $$
$y = x^2 + 1$ is a parabola with vertex at $(0,1)$ and axis vertical, and the horizontal line $y=2$ cuts the parabola at $(1,2)$ and $(-1,2)$.
How to find the line integral $\int_C P dx +Qdy$ ? Please suggest.
I can evaluate the line integral along $C_1:$ $y=2$ and unable to evaluate along the boundary of the parabola $C_2$.
Please help
So, I gathered that you wish to verify the validity of Green's theorem and that you are stuck on computing the line integral over $C_2$.
The parabola can be parametrized by $\gamma(t) = (t, t^2+1), t \in [-1,1]$. So, the line integral can be computed as
$$ \int_{-1}^1 1\cdot (\sin t - (t^2+1)^2) dt + \int_{-1}^1 2t \cdot (t-(1+t^2)^2)dt = \cdots = -\frac{12}{5}. $$
Since the integral over $C_1$ is 8, you get the intended result.