Let S be the part of the plane x + y+ z = 2 in the first octant oriented up, and let F(x, y, z) = (x + Y, x – y, 2z). If we change the surface integral (f. F · dS into iterated integrals, we get: 2-r 2 * (x + z)dydx o f S (4 – 2y)dydx O 13 *(4 – 2y)dydx 2-x OS S*(4 – 2y)dydx 2-x

Q: Evaluate the surface integral | f(x,y,z) dS using an explicit representation of the surface. S…

A: To evaluate the surface integral using explicit  representation of the surface of function…

Q: 5. Use the Divergence Theorem to calculate the surface integral SF dS, where F(x, y, z) = (xye , xy?…

A: By divergence theorem value of the surface integral is

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: Evaluate the surface integral. ) ر م(ل 2) ds S is the part of the cylinder x² + y2 = 16 that lies…

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Q: Evaluate the integral for y dx + z*dy + x²dz, where C is the triangular closed path joining the…

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Q: 5. (a) Evaluate the surface integral /3² dS where o is part of the cone z = VT + y² that lies…

A: As you post multiple questions According to the guideline I only solve the first problem please post…

Q: Consider the solid Q bounded by the surface of equation y = x2 + z2 and the planes y = 4, z/3 = x, z…

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Q: Use the Divergence Theorem to calculate the surface integral ffs F- dS where F(x,y, z)=…

A: The given function: F(x,y,z)=sin(y+z)i+(4x2y+8y3)j+20x2zk Use Divergence theorem, to find the…

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

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Q: 3) Evaluate the surface integral. I| (x²z+ y²z) ds where S is the part of the plane z = 4 + x +y…

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Q: 2. Evaluate the surface integral f,yz ds ; o is the part of the plane x + y + z = 2 that lies in the…

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Q: Evaluate the surface integral fs F. dS where F = (4x, -2z, 2y) and S is the part of the sphere x² +…

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Q: Evaluate the surface integral ff G (x, y, z) dS, C where G (x, y, z) = x+y+z, and C is the…

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Q: Q#5. Evaluate the surface integral ff(x,y,z)dS When f(x, y, z)= 3xyz and is the portion of the plane…

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Q: 4. Find the volume V of the solid S that is bounded by the elliptic paraboloid defined by 2x2 + y2…

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Q: Evaluate the surface integral. || x²yz ds, s is the part of the plane z = 1 + 2x + 3y that lies…

A: To find ∬Sx2yz dS Here 0≤x≤5  and 0≤y≤2 S→=xi→+ yj→+…

Q: 7. Use Stokes's Theorem to write S.(2y² + 6x*)dx + (2x² + 7y³)dy+ (2z° – yz)dz as a surface integral…

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Q: 5. Evaluate the surface integral S_(x+ y) ds ; o is the part of the plane 2x + y +z = 3 that lies in…

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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: 4. Use the divergence theorem to evaluate the surface integral F. dS where F = xzi + yzj + z²k and S…

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Q: 15- Evaluate the surface integral f, xy dswhere x is the part of the plaue y = 1-x-z that lies in…

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Q: Evaluate the surface integral. x² + y2 + z• ds S is the part of the cylinder x2 + y2 = 25 that lies…

A: Given, Where, S is the part of the cylinder x2 + y2 = 25 which lies between the planes z = 0 and z =…

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: 5. Use our Early Parameterization step by step including sketch of the surface to evaluate the…

A: Given that The equation of cylinder in the xz-plane as x2+z2=1 Also, the given surface integral lies…

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: Evaluate , yz+4xyds where S is the surface of the solid bounded by 4x+2y+z=84, z-0, y=0 and x-0.…

A: From the given surface equation 4x+2y+z=84, find z. Then differentiate partially with respect to x…

Q: Evaluate double integral f.ns where F=xi-yj+(z2-1)k and s is closed surface bounded by the planes…

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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…

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Q: Evaluate the surface integral. y ds S is the surface z = 3/2 0 <x < 4, 0 < y< 3 +

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Q: Let S be the negatively oriented surface defined by R(u, v) = (u², v², uv), ere (u, v) € [0, 1] ×…

A: This is a problem of vector calculus.

Q: 4. Find the volume of the solid bounded by the plane z = 0 and the paraboloid z=1-x² - y². (if using…

A: Solution :-

Q: 2- Use Gauss's divergence theorem to evaluate the surface integral / ry dydz + 2y°dxdz + y?zdxdy),…

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Q: Let R be the triangular region in the xy-plane with corner points (0,0). (1,0) and (0,1). and let…

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Q: Use Stokes's Theorem to calculate the surface integral S Ss F - dS where F(x, Y, z) = xi + (2ze² –…

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Q: #3. Evaluate the surface integral xz ds, where S is the part of the plane 2x +2y +z = 4 that lies in…

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Q: 4) Evaluate the surface integral [[F•dS where F(x,y,z)= (yz,-xz,z)and S is the piece of the…

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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: Let S be the negatively oriented surface defined by R(u, v) = (u², -uv, v²), where (u, v) € [0.2] ×…

A: Given Ru,v=u2,-uv,v2 and Fx,y,z=y,-xz,y Now, Ruu,v=2u,-v,0Rvu,v=0,-u,2v So,…

Q: 10. Let S be the surface of the solid bounded by the paraboloid z= the plane z=4 as in the picture.…

A: Given: x² + y² = z and z = 4

Q: E is the solid in R3 bounded by the cone surface z = square root (x2+ y2 ) and the plane z = 3, and…

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Q: Set up and evaluate the surface integral ||(4x+z+6)dS over the surface S ř (u,v)=(v, 4cos u, 4sin u)…

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Q: Evaluate the surface integral. JI (12 + y2 + 2) ds S is the part of the cylinder x2 + y2 = 9 that…

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Q: 8. Use the Divergence Theorem to calculate the surface integral for F(x, y, z) = (x³ +y³)i+ (y³ +…

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Q: Use Stockes' theorem to calculate the line integral fe F.dr where F (2.52) = (szy + e)i + (2 – Iny)3…

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Q: Evaluate the surface integral fSs x dS, where S is the cylinder y²+z² = 4 between (2) the planes x =…

A: We have to evaluate the surface integral

Q: b) Use the Gauss-divergence Theorem to find the surface integral ff, F.n ds, where S is cylinder x²…

A: As stated in question we will use Gauss-divergence Theorem to solve this question.

Q: 1. Evaluate the surface integral SL xz² ds ; o is the part of the cone z = Vx2 + y² that lies…

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Q: 4. Evaluate S. F di where F(x, y, z) = (x – y²)i+ (y – z²)j+ (z – x²)k and C is positively oriented…

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Q: Evaluate the surface integral ff G (x, y, z) dS, C where G (x, y, z) = x+y+z, and C is the…

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Q: Integrate G(x, y, z) = x + y + z over the surface of the cube cut from the first octant by the…

A: Definition used:

Domain

In simple mathematics terms, a domain is defined as the space on which all the possible inputs for the function will lie. It can also be expressed as a set of departures for the required function.