A parametrization of the portion of a surface z=x²+y2 below the plane z= 16 is given by 7(u,v)= [ucos(v), usin(v), u²], 0 su s4; 0svs2T. Select one: O True O False

Q: 1. Prove that fo r*dr + rydy = where C1 is the line segment from (0,0) to (1,0) by using…

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Q: 2. Prove that fo r*dr + xydy = -+- where C2 is the line segment from (1,0) to (0, 1) by using…

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Q: Evaluate the surface integral. x dS, S is the part of the plane 16x + 8y + z = 16 that lies in the…

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Q: Evaluate the surface integral. xz ds, S is the part of the plane 2x + 2y + z = 4 that lies in the…

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Q: Q3// A surface S is defined by 2x2 + z = 1 and the bounded by the planes y = 0,y 2, x 0,z = 0 in the…

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Q: Calculate F· dS where F = (4x*z, 4y°z, 3:*) S is the surface of the solid bounded by the hemispheres…

A: Given

Q: Q2// A surface S is defined by x2 + z = 9 and the bounded by the planes y = 0, y = 1,x = 0,z = 0 in…

A: Given that V=xy2z And y=0, y=1,x=0,z=0 The given surface is x2+z=9 The integral formed is,…

Q: use a parametrization to find the flux ∬S F . n dδacross the surface in the specified direction. F =…

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Q: Calculate the surface integralff, ds, where S is the hemisphere x² + y² + z² = 1, z ≥ 0. The…

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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: Give a parametrization, including the appropriate u- and v-bounds, for the portion a cylinder of…

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Q: Consider x=h(y,z) as a parametrized surface in the natural way. Write the equation of the tangent…

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Q: Evaluate the surface integral xyzdS where S is the part of the plane z = 1 + 2x + 3y that lies above…

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Q: Use Stokes' Theorem to caleulate ſ. F - dř, where F = zyi+3=3+5zk, and C is the closed path around a…

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Q: Compute the surface area of the surface segment parametrised by x(u, v) = u² cos v, y(u, v) = u² sin…

A: Given, xu,v=u2cosv,yu,v=u2sinv,zu,v=u2where 1≤u≤3 and 0≤v≤2π To find the surface area.

Q: Find the value of the integral fs (V x A). da if the vector A = yi + zj + xk and Sis the surface…

A: Given the vector A=yi+zj+xk and S is the surface defined by the paraboloid z=1-x2-y2

Q: Evaluate the surface integral. /| x*yz ds, s is the part of the plane z=1+ 2x + 3y that lies above…

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Q: Find the parametrization r(t) = [x(t), y(t), z(t)] of the curve obtained by intersecting the…

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Q: Miscellaneous volumes Use a triple integral to compute thevolume of the following regions.

A: The given equation of cone is and plane is, z=1-x2+y2 x+y+z=1

Q: Evaluate the surface integral  ffR(x+y)dS where σ is the portion of the plane z = 6 − 2x − 3y in the…

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Q: A solid G is bounded by the paraboloid z = 4 – x2 – y², the cylinder 22 + y? = 4 and the plane z =…

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Q: Use Stokes' theorem to calculate JJ (A) nds by A = 3yi-xzi+yz²k where S is the surface of the…

A: The given vector field is: A=3yi-xzj +yz2k The surface S is the paraboloid 2z=x2+y2 bounded by z=2.…

Q: Match the given parametrizations to the surfaces shown in the figures. Answer Bank u cos (v) ((u² +…

A: The complete solutions are given below

Q: Evaluate the surface integral. /| xz ds, S is the part of the plane 2x + 2y + z= 4 that lies in the…

A: We need to find surface integral.

Q: Use Stokes' Theorem to evaluate SSs curlF · dS, where F(x, y, z) = 7xyi + 3yzj+4z²k and S is the…

A: Consider the provided question, Use Stokes' Theorem to evaluate ∫∫Scurl F·dS,where F(x, y, z)…

Q: use a parametrization to find the flux ∬S F . n dδacross the surface in the specified direction. F =…

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

Q: use a parametrization to find the flux ∬S F . n dδacross the surface in the specified direction. F =…

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Q: b) Use Gauss theorem to evaluate F.ndS, where F(r, y, 2) = (2x²y² + In z) i – xy³ j – tanh(a² + y³)…

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Q: Calculate F. dS where F = (4x°z, 4y*z, 32*) S is the surface of the solid bounded by the hemispheres…

A: Given that F→=(4x3z, 4y3z,3z4) and S is bounded by z=16-x2-y2, z=1-x2-y2, z=0 Now,…

Q: Use a parametrization to find the flux F•n do across the surface in the given direction. F= - xi -…

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Q: If E is the solid shape bounded by x 20,y 20,z 20 and the plane 6x + 4y + 3z = 12 evaluate %3D (y +…

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Q: Find the flux of F(x, y, z) = (z, 4x, y) through the surface S, when S is the part of the plane 2x +…

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Q: Evaluate the surface integral (4x² + 3y)dS where S is the part of the plane 2.x + 5y + z = 48 inside…

A: Surface integral is a general form of multiple integrals over a surface. Sometimes it is same as…

Q: Use Green's Theorem to calculate the work done by the force F on a particle that is moving…

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Q: Calculate F. dS where = (4e°z, 4y°z, 324) F: S is the surface of the solid bounded by the…

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Q: A: Determine the mass and center of mass of the solid that bounded by the paraboloid z = 9y? + 9x?…

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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: use a parametrization to find the flux ∬S F . n dδacross the surface in the specified direction. F =…

A: Given surface, F=2xyi+2yzj2xzk           ...(1) Given plane, x+y+z=2a                ...(2) Where…

Q: Calculate F. dS where F = (42*z, 4y°z, 3z*) /25 – x² – y², z : 16 — г? — у? = Z 2 S is the surface…

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Q: 9. Let S be the surface parametrized by (u, v³, u + v) where 0 <u <1 and 0 < v<1. Calculate y dS.

A: The solution are next step

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

A: Strokes theorem:

Q: Evaluate the surface integral (4x² + 3y)dS where S is the part of the plane 2.x + 5y + z = 48 inside…

A: We need to calculate:∬S4x2+3ydS over the surface S,which is the part of plane 2x+5y+z=48, inside the…

Q: Use Stokes' theorem to calculate (V × A) · ndS with A = 3yi – xzj+ yz²k, where S is the surface of…

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Q: Use Stokes’ Theorem to evaluate F(x, y, z) = 3zi + 4x j + 2yk; C is the boundary of the paraboloid…

A: Use Stokes theorem

Q: a) Evaluate (-x i - yj+ 2z k) - n dS, where o is the portion of the paraboloid z = 3r2 + 3y? between…

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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: Find SS F dS where F(x, y, z) = z i+(3x – 1)j+ yk and S is the portion of the plane x + y + z = 1…

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Q: Use Stoke’s Theorem to evaluate the line integral ∮C(z2−y2)dx +(x2−z2)dy +(y2−x2)dz, where the curve…

A: Given equation of the curve is ∮C(z2-y2)dx+x2-z2dy+(y2-x2)dz ,the equation of the paraboloid…