An area $R$ in the $xy$-plane is bounded by two curves $ C_1 $ and $ C_2 $. The two curves form a closed curve $C$ where the positive direction of rotation is anti-clockwise. The two curves can be described with the functions:
$ C_1:y=x $
$ C_2:y=\sqrt{x} $
We also have the vector field: $ F(x,y) = 3xy^2i+2yj $
I am supposed to find the line integral over $ C_2 $ by using Green's theorem
My soultion so far is using a combination of Green's theorem and normal line integral, like this:
$$ \int_{C_2} F\cdot dr = \int_{C_1+C_2} F\cdot dr - \int_{C_1} F\cdot dr $$
By using Green's theorem on the $ C_2 + C_1 $ integral we get:
$$ \int_{C_1+C_2} F\cdot dr = $$ $$ \int \int_D \frac{\partial F_2}{\partial x } - \frac{\partial F_1}{\partial y } da = $$ $$ \int_0^1 \int_x^\sqrt{x} (-6xy) dydx = -\frac{1}{4} $$
By completing the integral over $ C_1 $ we get:
$$ \int_{C_1} F\cdot dr $$ $$ \int_0^1 (F(x(t), y(t)) \cdot r'(t)) dt $$ Where $ r(t) = ti + tj \Rightarrow r'(t) = i + j $ $$ \int_0^1 (3t^3+2t) dt = \frac{7}{4} $$
By subtracting the two terms we get $ - \frac{1}{4} - \frac{7}{4} = -2 $ Which is almost the right answer?
If we check by using normal line integration over $ C_2 $ we get: $$ \int_{C_2} F\cdot dr $$ $$ \int_0^1 (F(x(t), y(t)) \cdot r'(t)) dt $$ Where $ r(t) = t^2i + tj \Rightarrow r'(t) = 2ti + j $ $$ \int_0^1 (6t^5+2t) dt = 2 $$
I suspect I have an integral going in the wrong direction or something like that. There might be an easier method to do this, or maybe I am completely wrong. Any help is appreciated.
EDIT Its seems like all of my calculations were right except the boundaries on the line integral of $ C_2 $ when checking for the correct answer. Since the direction of the closed curve $ C $ is anti-clockwise, we need to find the line integral of $ C_2 $ from $ 1 \rightarrow 0 $. Doing this we get: $$ \int_{C_2} F\cdot dr $$ $$ \int_1^0 (F(x(t), y(t)) \cdot r'(t)) dt $$ Where $ r(t) = t^2i + tj \Rightarrow r'(t) = 2ti + j $ $$ \int_1^0 (6t^5+2t) dt = -2 $$ Which corresponds with the answer we got using Green's theorem.