Use Green’s Theorem to evaluate the line integral along the given positively oriented curve. integral through C y^4dx+2xy^3dy, C is the ellipse x^2+2y^2=2

Use Green’s Theorem to evaluate the line integral. ∫_c (e^-x²/2 - y) dx + (e^-y²/2 + x) dy C: boundary of the region lying between the graphs of the circle x = 6 cos θ, y = 6 sin θ and the ellipse x = 3 cos θ, y = 2 sin θ

Use Green’s Theorem to evaluate the line integral. ∫_c 2xy dx + (x+y) dy C: boundary of the region lying between the graphs of y = 0 and y = 1 - x²

Use Green’s Theorem to evaluate the line integral along the given positively oriented curve.

$$∫c (y + e^√x)dx+(2x+cosy^2)dy,$$

C is the boundary of the region enclosed by the parabolas

$$y = x^2 and x = y^2$$

Use Green’s Theorem to evaluate the line integral along the given positively oriented curve. integral C $y^3dx-x^3dy$, C is the circle $x^2+y^2=4$

Use Green's Theorem to evaluate the line integral along the given positively oriented curve. $\int_{C} \cos y d x+x^{2} \sin y d y$ C is the rectangle with vertices $(0,0),(5,0),(5,2)$ and $(0,2)$

Use Green’s Theorem to evaluate the line integral along the given positively oriented curve. $\int_{C} \sin y\ d x+x \cos y\ d y, \quad C \text { is the ellipse } x^{2}+x y+y^{2}=1$

Find a function f such that F=delf F(x,y)=(1+xy)e^xyi+x^2*e^xyj, C: r(t)=costi+2sintj, 0<=t<=pi/2

Use Green’s Theorem to evaluate integral C F.dx (Check the orientation of the curve before applying the theorem.) F(x,y)=<ycosx-xysinx,xy+xcosx>, C is the triangle from (0, 0) to (0, 4) to (2, 0) to (0, 0)

Evaluate the line integral by using Green’s Theorem. closed integral through C xydx+x^2y^3dy C is the triangle with vertices (0, 0), (1, 0) and (1, 2)

Use the Divergence Theorem to evaluate

$$∫∫s(2x + 2y + z^2)dS$$

where S is the sphere

$$x^2+y^2+z^2=1$$

Use Green’s Theorem to evaluate the integral ∫_c (y-x) dx + (2x - y) dy for the given path. C: boundary of the region lying between the graphs of y = x and y = x² - 2x

Evaluate the line integral along the curve C.

$$\int _ { C } ( y - x ) d x + x y d y$$

C: the line segment from (3, 4) to (2, 1)

Find dy/dx if x^2 + y^2 = -2xy.

(A) 1

(B) -1

(C) (x-y)/(x+y)

(D) (x+y)/(x-y)

(E) (x+2y)/x