1. (20pts) evaluate ∫_c (x²)dx+(x)dy, where c is the circle given by x² + y² = 4 oriented counterclockwise.

Explanation:

Step1: Use Green's Theorem

Green's Theorem states that $\oint_{C} Pdx + Qdy=\iint_{D}(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})dA$, where $P = x^{2}$, $Q = x$, and $D$ is the region bounded by the closed - curve $C$. First, find the partial derivatives: $\frac{\partial Q}{\partial x}=1$ and $\frac{\partial P}{\partial y}=0$.

Step2: Set up the double - integral

Then, $\oint_{C}x^{2}dx + xdy=\iint_{D}(1 - 0)dA=\iint_{D}dA$. The region $D$ is a disk of radius $r = 2$ since the equation of the circle is $x^{2}+y^{2}=4$.

Step3: Evaluate the double - integral

The double - integral $\iint_{D}dA$ represents the area of the region $D$. The area of a disk with radius $r$ is $A=\pi r^{2}$. Here, $r = 2$, so $\iint_{D}dA=\pi(2)^{2}=4\pi$.

Answer:

$4\pi$