Find the circulation of F= (xy)i+(yz')j-(2xyz)k around the curve C in the counterclockwise direction (when viewed from above), where C is the portion of the plane 3x+3y+z=3 in the first octant.

Q: Find the coordinates of the centroid of the curve given belov x = cos t+tsin t, y =t cos t- sin t,…

A: Given data: The x-coordinate of the curve is x=cos t+tsin t. The y-coordinate of the curve is y=tcos…

Q: find the outward flux of F across the boundary of D. F =-2x i - 3y j + z k D: The upper region cut…

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Q: Evaluate the unoriented line integral (22 + y° + z²) ds (2x² along the parametric curve a : [0, 1]…

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Q: Evaluate the line integral f e-*(cos y dx - sin y dy), where c is the rectangle with vertices (0,0),…

A: The line integral function is ∫Ce-xcosydx-sinydy. The vertices of the rectangle is 0, 0, π, 0, π,…

Q: Integrate f(x,y) = √(64 - x2) over the smaller sector cut from the disk x2 + y2 ≤ = 64 by the rays θ…

A: Concept:   The calculus helps in understanding the changes between values that are related by a…

Q: A point moves along the intersection of the elliptic paraboloid z = 9x? + 3y² and the plane.x…

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Q: Calculate F - dS where F (e* sin y, e² cos y, yz²) S is the surface of the box bounded by the planes…

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Q: Find the circulation of F(x, y, z) = (-y sin 2, x sin z, xy cosz +12) around the circle cut from the…

A: Given that,F(x, y, z)=(-ysinz, xsinz, xycosz+12) around the circle cut from the sphere x2+y2+z2=5 by…

Q: Compute the flux of F = 3(x + z)i +j+3zk through the surface S given by y = x2 + z2, with 0 0, z >…

A: From given data we see that the surface is oriented towards xz plane,  Hence,  d→s=dxdzj→

Q: Find the polar moment of inertia about the origin of a thin triangular plate of constant density d =…

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Q: use the Divergence Theorem to find the outward flux of F across the boundary of the region F = x2i -…

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Q: Find the flux of F (x, y, z) = 4.xi+ 4yj+ 2k outward (away from the z-axis) through the surface cut…

A: The field lines of a vector field F through surfaces with unit normal n, the angle from n to F is θ.…

Q: Find F. dr, where F(x, y) = (xy² + cos x)i+(x²y+sin y)j and C is the ellipse C x2 + 2y2 = 9,…

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Q: Use cylindrical coordinates to evaluate the triple integral □ لاا √² + y²dV, where E is the solid…

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Q: Find the coordinates of the centroid of the curve x = cos t, y = t + sin t, 0<=t <= pai.

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Q: 2. Evaluate f, z³ /x² + y² + z² dV where E is the solid hemisphere that lies above the xy-plane and…

A: Solution:- equation of sphere having centre origin (0,0) and radius=1 x2+y2+z2=1…

Q: Find the centroid (δ = 1) of the solid enclosed by the cylinder x2 + y2 = 4, bounded above by the…

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Q: Evaluate the line integral I of the one-form w = xy dx + 12y dy over the parametric curve a : [0,…

A: Given: ω=xydx+12ydy To find: I=∫αω  where α:[0,ln2]→ℝ2 with α(t)=(e2t,e-2t) ∴x=e2t,y=e-2t Now,…

Q: Find the circulation of F= (y +z)i+(x +z?)j+(x² + y² )k around the curve C in the counterclockwise…

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Q: Calculate F. dS where F = (e" sin y, e² cos y, yz²) S is the surface of the box bounded by the…

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Q: Which one of the following is the parametric equa- tions of the line that is normal to the surface…

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Q: Determine the centroid, C(X,9.2). of the solid formed in the first octant bounded by z+y- 16 = 0 and…

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Q: 3. Calculate the rotational flow of F(x, y, z) = -j + xj + x²k, that goes through the cone given by…

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Q: Calculate F. dS where F = (e* sin y, e² cos y, yz²) S is the surface of the box bounded by the…

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Q: Given the scalar function f(x,y)=2xy,calculate the line integral along the ellipse curve from point…

A: Given data:- The given curve is, x29+y24=1 The given scaler function is, f(x,y)=2xy t E 0,π/2

Q: Calculate F- dS where F = (e" sin y, e* cos y, yz²) S is the surface of the box bounded by the…

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Q: Evaluate the unoriented line integral (2æ² + y² + z²) ds along the parametric curve a : [0, 1] → R³…

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Q: Find the counterclockwise circulation and the outward flux of thefield F = (-sin y)i + (x cos y)j…

A: Use tangential form of Green's Theorem to find the counterclockwise circulation. Counterclockwise…

Q: Green’s Theorem for line integrals Use either form of Green’sTheorem to evaluate the following line…

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Q: Exercise (2): Calculate fe xydx + 3y*dy where C is the curve in xy plane, x = y² from (0,0). to…

A: we have to following integral

Q: B. Compute the shared mass between the polar curves (r = 1+ cose) and (r = 1- cose), usę density…

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Q: Integrate F =-(y sin z)i + (x sin z)j + (xy cos z)k around the circle cut from the sphere x2 + y2 +…

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Q: Find the work done in moving a point charge Q- - 20 1C from.the origin to (4,0,0) m in the ficld E-+…

A: To find - Find the work done in moving a point charge Q = -20 μC from the origin to 4, 0, 0m in the…

Q: Evaluate the circulation of G = ryi + zj + 2yk around a square of side 7. centered at the origin,…

A: Given  G=xyi+zj+2yk around a square of side 7

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 average height of the hemispherical surface z = √(a2 - x2 - y2) above the disk x2 + y2 ≤ a2…

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Q: Find the work done by F in moving a particle once counterclockwise around the given curve = (2x –…

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Q: Find the centroid of the solid bounded by the parabolic cylinder 1- y? and the three planes given…

A: To find the centroid of the solid bounded by the parabolic cylinder: The region is defined as…

Q: Compute the scalar line integral xy ds where C is the part of the circle of radius 3, centered at…

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Q: Evaluate the circulation of Gayi + zj+5yk around a square of side 9, centered at the origin, lying…

A: Solution :

Q: Find the flux of F(x, y, 2) = (r, y, 2) across the surface o which is the surface of the solid…

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Q: find ⌠⌠F*dS if F(x,y,z)=2xi+2yj+2zk and S is the paraboloid z=8-x^2-y^2 lying above the rectangle…

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Q: Evaluate the unoriented line integral +y² + z²) ds along the parametric curve a : [0, 1] → R³ with…

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Q: C is the intersection curve between the paraboloid z = x² + y? and the plane 6x + 8y – z = 21,…

A: Strokes theorem:

Q: Find the flux of F across S, where F(x, y, z) = xyi + y²j + zk and S is the part of the paraboloid z…

A: According to question given that F(x,yz,)=xyi^ + y2 j ^+z k^S is the part of the paraboloid z=x2+y2,…

Q: Let F = (x'e* +2.xy - 2y)i+(cos y+x² +3x)j and let C be the ellipse with equation (2x - 2y) +(x+ y)°…

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Q: Find an equation for the plane through the origin such that the circulation of the flow field F = zi…

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Q: Find the coordinates of the centroid of the curve x = cos t + t sin t, y = sin t - t cos t, 0…

A: Given,

Q: Compute the line integral [(yze²v² + y² + 2 − 2x) dx + (xze²v² + 2xy) dy + (xye²³² +x+32²) dz, where…

A: We have to compute the line integral ∫Cyzexyz+y2+z−2xdx+xzexyz+2xydy+xyexyz+x+3z2dz, where C is the…

Q: Calculate the circulation of the flow v=-8xy i+4y j around the triangle with vertices (0, 0), (3,…

A: According to our policy, we are supposed to answer the first question (question2). By using the…