Let C be the positively oriented square with vertices (0,0), (3,0), (3,3), (0,3). Use Green's Theorem to evaluate the line integral ∫c 7y^2x dx+6x^2y dy.

Q: Find a parametrization of the vertical line passing through the point (9, 5, 0) using t = z as a…

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Q: Find the centroid of the “triangular” region bounded by the lines x = 2, y = 2 and the hyperbola xy…

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Q: Determine the solid bounded by the surfaces 9x2+4y2=30 and the plane 9x+4y−6z=0 in the first octant.

A: Given surface is 9x2+4y2=30 and plane is  9x+4y-6z=0 Bounded region in first octane ⇒x≥0, y≥0, z≥0…

Q: Find a point on the hyperboloid x2 + 4y2 – z2 = 1 where the tangent plane is parallel to the plane x…

A: Given  and plane

Q: Determine the centroid bounded by the parabolas x=y^2 and x^2=-8y

A: Consider the given curves. x2=-8yx=y2

Q: Verify Green's theorem in a plane with respect to S. [(x2 - y²)dx + 2xy dy], where C is the boundary…

A: Using greens theorem

Q: Compute the moments of the semicircle x2 + y2 = R2, y ≥ 0 as line integrals. Verify that the…

A: Given semicircle x2+y2=R2,  y≥0 The figure of this semicircle is shown below. In the above figure,…

Q: Let C be the positively oriented square with vertices (0,0) (2,0), (2,2), (0,2) Use Green's Theorem…

A: Consider the provider question,

Q: Classify all critical points for the surface flaiy)= 2x-y-2xy.

A: We have to classify all the critical points for the surface…

Q: Use Stoke's theorem to evaluate the integral xydx +2x 'dy-&r 'zdz where C is the boundary of the…

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Q: IntegrateF(x, y, z) = z, over the portion of the plane x + y + z = 4 that lies above the square…

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Q: Use Stokes' Theorem to find the line integral of F = 3yi + 2zj + 5xk around the triangle obtained by…

A: Given function is F→=3yi^+2zj^+5xk^. We have to evaluate the line integral of F→ around the triangle…

Q: Use Green's theorem to evaluate $(xy + 3x²)dx + (3y – 2x)dy, along C, where Cis the triangle with…

A: Given Integral ∮(xy+3x2)dx +(3y-2x)dy along c where c is the triangle with vertices (0, 0), (0, 3)…

Q: Integrate G(x, y, z) = xyz over the triangular surface with vertices (1,0, 0), (0, 2, 0), and (0, 1,…

A: To Determine: integrate Gx,y,z=xyz over the triangular surface with vertices (1,0,0),(0,2,0) and…

Q: Find the centroid of a thin, flat plate covering the region enclosed by the parabola y2 = x and the…

A: Given information:

Q: Calculate the scalar line Integral where the function is f(x,y)= 1 / x²+y²+1 and the curve is the…

A: We have given function fx,y=1x2+y2+1 and the curve joining the point 0,0 1,0 1,1 0,1. Let A0,0,…

Q: the normal line of the surface [z = (4x^2 + y^2] at the point (1,1,5) O 8x-2y+z=5 O 8x+2y+z=5…

A: We have to find the normal line of the surface.

Q: A _____ is the set of points (x, y) in a plane such that the difference in distances between (x, y)…

A: The standard form of the equation of the hyperbola is,

Q: Find the normal vector to the surface z = xe™Y at the point (2,0, 2).

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Q: /Sketch the Surface Z=V9-X+y- and verify that fxg = Exx %31

A: Second order Partial Differentiation

Q: Integrate F = 3x2yi + (x3 + 1)j + 9z2k around the circle cut from the sphere x2 + y2 + z2 = 9 by the…

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Q: | Let C be the positively oriented square with vertices (0, 0), (2, 0), (2, 2). (0, 2). Use Green's…

A: greens theorem

Q: Let C be the positively oriented square with vertices (0, 0), (2,0), (2, 2), (0, 2). Use Green's…

A: Let the given square be OABC which is positively oriented.

Q: find the points on the surface + z² = 1 that have their tangent plane perpendicular to the vector…

A: For tangent plane of the surface f(x,y,z)=k at a given point normal vector to tangent plane  will be…

Q: Find the centroid of a thin, flat plate covering the region enclosed by the parabolas y = 2x2 and y…

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Q: A point moves along the intersection of the elliptic paraboloid z = 9x + 7y and the planex = 1. At…

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Q: Let C be the positively oriented square with vertices (0, 0), (2,0). (2, 2). (0, 2). Use Green's…

A: Green's Theorem:  For a positively oriented curve C∫CP dx+Q dy=∫∫D∂Q∂x-∂P∂y dAfor the region D…

Q: What is the surface area of the portion of the surface z=x2+y that lays over the triangle on xy-…

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Q: Find a vector normal to the surface x2 + yz = 5 at (2, 1, 1).

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Q: Find the center of mass of a thin triangular plate bounded by the y-axis and the lines y = x and y =…

A: Given problem is :

Q: The normal vector to the tangent plane for the surface xy² z³ 2 3 = -4, at the point (1, 2, -1) is:…

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Q: Find the distance from the point (1, 1, 1) to the plane 2x + 2y + z = 0.

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Q: A point moves along the intersection of the elliptic paraboloid z = 4x + 11y and the plane y = 4. At…

A: given information is z=4x2+11y2 The plane y=4To find :Rate of change of z with respect to x when the…

Q: Find the orthogonal trajectories of the conicoid (x + y)z = 1 of the conics in which it is cut by…

A: Given:The equation of conicoid is (x+y)z=1The system of the plane is x-y+z=K, where k is…

Q: ) Find the points on the ellipsoid 6x² + y² + 3z² = 4 where the tangent plane is parallel to the…

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Q: use divergence theorem to find the outward flux of f =2xzi-3xyj-z^2k across the boundary of the…

A: By divergence theorem

Q: (b) Integrate (x, y,z)=(y+z)+1 the paraboloid 4x - y-z -0 by the plane x 1. over the surface cut…

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Q: Find the area of the portion of the paraboloid x = 4 - y2 - z2 that lies above the ring 1<=y2 +…

A: Given surface is  The required surface area is given by where

Q: 2. Calculate the centroid of the thin plate R which bounded by r = 1, y = 0 and, 2² + y² = 4.

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Q: Find the equation of the tangent plane to the surface 3D1 16 at the point (1,-3,2) O a. 2x+4y-3z=12…

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Q: Find the tangent plane to the elliptic paraboloid z = x² – 2y at the point (1,1,1). - O A. z = 2x –…

A: Given that: z=x2-2y2      and point is:   (1, 1, 1)

Q: Verify Stokes' Theorum for F(x,y,z) = (y,-x,yz) if S is the paraboloid z = x2 + y2 with x2 + y2 = 1,…

A: Given  Fx,y,z=y,-x,yz Then, F→=yi^-xj^+yzk^

Q: Find the centroid of a thin, flat plate covering the region enclosed by the x-axis, the lines x = 2…

A: Given information:

Q: (a) Integrate g(x,y,z) = Inx²+y over the surface of the region cut from the plane 4x+3y+2z=1 by the…

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Q: Evaluate the double integral / ydrdy where T is the triangle with vertices (0,-1), (0, 5), (4, 3).

A: First, draw the region of integration. Then, use the region to setup the double integral.

Q: Find the line integral of z2 dx + y dy + 4y dz, where C consists of two parts, C, and C,. C, is the…

A: Given line integral ∫Cz2dx+ydy+4ydz, where C consists of two parts C1 and C2. C1 is the intersection…

Q: Evaluate the line integral 3xy ds where Cis line Segment Joining (-5,7) to (3,1) the

A: Introduction: A line integral in calculus is an integral where the function being integrated is…

Q: Find the centroid when revolved in y-axis: y³ = x² ; y = 4.

A: As per the question we are given a solid of revolution obtained by rotating the area between y3 = x2…

Q: Find the points on the surface xy – z2 = 1 that are closed to the origin.

A: Given surface is xy-z2=1. let xy-z2-1=0⇒z2-xy+1=0 Now distance from origin is…

Q: OFind the orthogonal Tradectories of the families of y2-5ax 42 - %3D where a being a Parameto

A: For find the orthogonal trajectory replacement of dy/dx with (-dx/dy) is required.