Evaluate \(\displaystyle{ \oint_{\mathcal{C}}{ (y^2 ~dx + 3xy ~dy) } }\) where \( \mathcal{C} \) is the boundary of the semiannular region D in the upper half plane between the circles \( x^2 + y^2 = 1 \) and \( x^2 + y^2 = 4 \).

Calculate the outward flux of \( \vec{F} = -x\hat{i} + 2y\hat{j} \) over the square with corners \( (\pm 1, \pm 1) \) where the unit normal is outward-pointing.

Evaluate \(\displaystyle{ \oint_{\mathcal{C}}{ y^2~dx+x^2~dy } }\) where \( \mathcal{C} \) is the triangle bounded by the lines \( x=0, x+y=1, y=0 \)

Let \( \mathcal{C} \) be the circle \( x^2 + y^2 = 4. \) Evaluate \(\displaystyle{ \oint_{\mathcal{C}}{ \left[ (3y-e^{\tan^{-1}x})~dx + (7x + \sqrt{y^4+1}) ~dy \right] } }\)

Evaluate \(\displaystyle{ \int_{\mathcal{C}}{ x~dx - x^2y^2~dy } }\) where \( \mathcal{C} \) is the triangle with vertices \((0,0)\), \((0,1)\) and \((1,1)\).

Evaluate \(\displaystyle{ \oint_{\mathcal{C}}{ P~dx + Q~dy } }\) where \( \mathcal{C} \) is the square with corners \( (\pm 1, \pm 1) \); \( P(x,y) = x + y^2 \) and \( Q(x,y) = y + x^2 \).

Evaluate \(\displaystyle{ \oint_{\mathcal{C}}{ P~dx+Q~dy } }\) where \(\mathcal{C}\) is the triangle with corners \((0,0)\), \((1,1)\), \((2,0)\), \(P(x,y)=y+e^x\) and \(Q(x,y)=2x^2+\cos y\).

Evaluate \(\displaystyle{ \int_{\mathcal{C}}{\vec{F} \cdot d\vec{r}} }\) where \(\mathcal{C}\) is the circle \(x^2+y^2=4\) and \(\vec{F}=\langle -y, x \rangle\).

Evaluate \(\displaystyle{\int_{\mathcal{C}}{\vec{F} \cdot d\vec{r}}}\) where \(\mathcal{C}\) is along \(y=x^2/2\) and \(y=x\) and \(\vec{F}=\langle y^2, x^2 \rangle\).

Evaluate \(\displaystyle{\int_{\mathcal{C}}{\vec{F} \cdot d\vec{r}}}\) where \( \vec{F}=\langle y^3, 6xy^2 \rangle\) and \(\mathcal{C}\) is the curve with the 3 segments:
1. \( y=0, 0 \leq x \leq 4\), 2. \( x=4, 0 \leq y \leq 2\), 3. \( y=\sqrt{x}, 0 \leq x \leq 4\)

Evaluate \(\displaystyle{ \int_{\mathcal{C}}{\vec{F} \cdot d\vec{r}} }\) where \(\mathcal{C}\) is boundary between the two circles \(x^2+y^2=1\) and \(x^2+y^2=3\) that lies in the first quadrant and \( \vec{F} = \langle e^x+2xy, 4x^2+\sin y \rangle \).

Determine the flux of the field \( \vec{F} = \langle -x, -y \rangle \) across \(x^2+y^2=16\).

Calculate the outward flux of \( \vec{F} = -x\hat{i} + 2y\hat{j} \) over the square with corners \((\pm 1, \pm 1)\) where the unit normal is outward-pointing.

Calculate the outward flux of \( \vec{F} = -x\hat{i} + 2y\hat{j} \) over the square with corners \((\pm 1, \pm 1)\) where the unit normal is outward-pointing.

Evaluate \(\displaystyle{ \oint_{\mathcal{C}}{ 2xy~dx + (x^2+y^2)~dy } }\) where \(\mathcal{C}\) is \( 4x^2 + 9y^2 = 36 \).

Evaluate \(\displaystyle{ \oint_{\mathcal{C}}{ 2xy~dx + (x^2+y^2)~dy } }\) where \(\mathcal{C}\) is \( 4x^2 + 9y^2 = 36 \).

Evaluate \(\displaystyle{ \oint_{\mathcal{C}}{(x^2-y^2)~dx + 2xy~dy} }\) on the curve which bounds the region \(\mathcal{R}: \{(x,y): 0 \leq x \leq 1, 2x^2 \leq y \leq 2x \}\)

Evaluate \(\displaystyle{ \oint_{\mathcal{C}}{ 2y~dx-3x~dy } }\) on the unit circle.

Verify Green's theorem for the line integral \(\displaystyle{ \oint_{\mathcal{C}}{ xy~dx + x~dy } }\) where \(\mathcal{C}\) is the unit circle.