Using Green's theorem in the plane, evaluate $ (ry² dy - x²y* dx). where C is the triangle with vertices (0,0), (-2, 1) and (2, 1) (You may use horizontal cross-sections method to compute the double integral).
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- Double integrate under the hyperbolic paraboloid z=x2-y2 on the triangular footprint made by the x and y axes and the line y=-x+1? Thank youIntegrate ƒ(x, y) = 1/ (1 + x2 + y2 )2 over a. Triangular region The triangle with vertices (0, 0), (1, 0), and (1, sqrt(3)). b. First quadrant The first quadrant of the xy-plane.Find div F and curl F if F(x, y, z) = x²i - 6j+yzk. div F- = 7 x + y curl F zi X
- Use Green's Theorem to evaluate∮tan^-1(y)dx-(xy^)/(1+y^2) dy where C is the square with vertices (0, 0), (1, 0), (1, 1) and (0, 1) and oriented counterclockwise. A. -1 B. 2 C. 1 D. -2Find the centroid of the thin plate bounded by the graphs of g(x)=x and f(x) = x + 6. Use the equations shown below with 8 = 1 and M = area of the region covered by the plate. == b f a M Sx[f(x) - g(x)] dx b y=-=-=[r²(x)-g²(x)] dx The centroid of the thin plate is (x,y), where x = (Type integers or simplified fractions.) and y =Tangent of x?/3 + y2/3 + z2/3 = a²/3 surface at any point ( xo , Yo ,Zo ) Show that the sum of the squares of the intersecting axes of the plane is constant.
- Find div F and curl F if F(x, y, z) = xzºi + 6y²x²j + 6z²yk. div F= curl F=Integrate F = 3x2yi + (x3 + 1)j + 9z2k around the circle cut from the sphere x2 + y2 + z2 = 9 by the plane x = 2.Use Green's Theorem to find the work done by F(x,y)=x(3x+y)i+xy^2j in moving a particle from the origin along the x-axis to (2,0), then along the line segment to (0,2), and then back to the origin along the y-axis.
- Use cross products to find the area of the quadrilateral in the xy-plane defined by (0, 0), (1, −1), (3, 1), and (2, 4).Let F = 2(x + y)i + sin(y) 7. Find the line integral of Faround the perimeter of the rectangle with corners (4,0), (4,8), (-2, 8), (-2, 0), traversed in that order. line integral =Let F=2(x + y)7+4 sin(y). Find the line integral of around the perimeter of the rectangle with corners (3,0), (3, 6). (-3, 6), (-3,0), traversed in that order. line integral