4. Use the divergence theorem to evaluate the surface integral F dS where F - xzi + yzj + z?k and S is the closed surface of the hemisphere %3D x2 + y? + z² = 4,z> 0.

Q: 386. Use the divergence theorem to calculate surface integral F-dS, where F(x, Y, z) = xªi – a³z²j+…

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Q: Evaluate the surface integral F. ds where F = 5xyi +8x²j+ 5yzk and S is the surface z = ïe², 0 ≤ x ≤…

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Q: Verify the Green's theorem to evaluate the line integral fo(2y²dx + 3zdy), where C is the boundary…

A: As y=x and y=x2 Equating the two, we get  x2=x⇒x2-x=0⇒xx-1=0⇒x=0,1 The graph of the two curves is as…

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

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Q: Evaluate the surface integral (x - z) dS where S is the portion of the cylinder r +z? = 1 above the…

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Q: If R is the region bounded by the lines y = x,y=0,x= 1, evaluate the dxdy double integral S by…

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Q: Use Stokes' Theorem to evaluate F.dr for F=(xz,-y,x²y), where C is the boundary of the open surface,…

A: Given : vector field F=xz,-y,x2y and C is the boundary of the open surface, S that is bounded by…

Q: Find the flux ofF across the oriented surface S' by evaluating the surface integral F.î dS. F = zî +…

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Q: Use Green's theorem to evaluate the line integral frdx + (a - 20)dy where C is the boundary of the…

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Q: 7. Evaluate the line integral f, F· dr where F(x,y, z) = x³i – y²j+ zk and C is the x2 + 2xy + y²…

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Q: Evaluate the surface integral 2 = 1 which is in the first octant. ry dS, where S is the part of r+y+

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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: Compute the triple integral f(x,v.z)dxdydz where D is the sphere with center O and radius R and…

A: Given that ∭Df(x,y,z)dxdydz where f(x,y,z)=R2-(x2+y2+z2). By the spherical coordinates,…

Q: Evaluate the surface integral. z2 S is the part of the paraboloid x = y2 + z² given by 0 sx 5 4 1 V5…

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Q: Evaluate the triple integral edV where E is bounded by the parabolic cylinder z = 1 - y² and the…

A: Given ∭x8ey dV

Q: Compute the surface integral // xy dS, where S is the quarter cylinder x² + y² = || S 1; x, y > 0…

A: Given surface S is the quarter cylinder x2+y2=1, Where x,y≥0 between z=0 and z=4 We have to…

Q: Let S: x2 + y? = 9,0szS7 be the surface of a closed cyllnder bounding a volume T.F=Ly², x², z²J. Use…

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Q: Let S:x+z = 16, 0<y<3 be the surface of a closed cylinder bounding a volume T. F= [z, y, x] Use 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…

A: We need to find surface integral.

Q: Evaluate the surface integral F. dS, where F(x, y, z) = xi+Yj+z k is taken over the paraboloid z = 1…

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Q: * Evaluate the surface integral SS F - d§, where F = z7 + y²7 + z² and S is the boundary of the…

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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: Evaluate the triple integral. sssE (x − y) dV,where E is enclosed by the surfaces z = x2 − 1, z = 1…

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

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Q: Evaluate the surface integral SP ( – 2) // where S is the portion of the cylinder r2+22 = 1 above…

A: It is given that S is the portion of the cylinder x2+z2=1 above the xy plane between y=1 and y=2.

Q: Evaluate the surface integral. (z + x2y) ds S is the part of the cylinder y2 + z2 = 16 that lies…

A: Given ∫∫S(z+x2y)dS where S is the part of the cylinder y2+z2=16 that lies between the planes x=0 and…

Q: Calculate the surface integral S FdS, where S is cylinder r² + y = 1, 0 + yz, 3 =

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Q: Evaluate the surface integral. x2z2 dS S is the part of the cone z2 = x2 + y2 that lies between the…

A: Given: The surface integral is given as, S is the part of the cone z2-x2 + y2 that lies between the…

Q: Evaluate the triple integral. (x - y) dV, where E is enclosed by the surfaces z = x2 – 1, z = 1 –…

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Q: 5- Evaluate the surface integral SS,G(r)dA, where G = arctan(y/x) and S is the surface z = a² + y²,0…

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Q: Calculate the surface integral ffs F - dS for F(x, y, z) = (xª, –x³z surface bounding the region…

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Q: Evaluate the surface integral. ds 2 + y32), o s x s S is the surface z = x3/2 0sx < 1, 0 s ys2

A: Given,                        z = 23(x32+y32)and domain D : 0 ≤ x ≤ 1, 0 ≤ y ≤ 2.∂z∂x= 23(32 x) =…

Q: Evaluate the surface integral. z2 ds, S is the part of the paraboloid x = y2 + z? given by 0 < x < 1

A: Evaluate the surface integral.

Q: Verify the Divergence Theorem by evaluating the surface integral the triple integral for F(x, y, z)…

A: From Divergence theorem: ∯F·nds=∫∫∫div(F)dV   Given; F=x3i+y3j+z3k   So,…

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

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Q: Use Gauss's Theorem to find the surface integral OF •dS across the positively-oriented S sphere of…

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Q: Use the Divergence Theorem to calculate the surface integral f Ss F- dS where F(x, y, z)=x²yi+ry²j+…

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Q: Q4// A surface S is defined by x2 +z = 1 and the bounded by the planes y = 0,y = 2,x 0,z = 0 in the…

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Q: Let  S = {(x,y) | y2 + z2 = 4, 0 ≤ x ≤ 4} Evaluate the surface integral of f over the surface S f =…

A: Put value of z from given condition in function we find integral ∫-22∫04x+4-y2dxdy…

Q: Evaluate the surface integral f, F. dS where F = 3xyi + 9x²j+3yzk and S is the surface z = xe", 0≤x≤…

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Q: Evaluate || F:ñ dS where S is the surface bounded by z = a - Vx² + y² and the xy-plane including the…

A: The vector field is F→=xπ, yπ, 3z-2π. The surface is bounded by z=a-x2+y2 and xy-plane. The function…

Q: Let S be the hemisphere x2+y2+z2=4, with z≥0. Evaluate the surface integral   ∬S z2 dS

A: we have the hemisphere x2+y2+z2=4with z≥0    we have to evaluate the surface integral ∫∫sz2ds so, we…

Q: Let S:x+z? = 16,0sys3 be the surface of a closed cylinder bounding a volume T.F=[z?, y, x²]. Use the…

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Q: Evaluate the line integral S.F · dr where F(x,y, z) = x³i – y²j+ zk and C is the poundary of the…

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Q: Evaluate [F d$ for F = yi+=j+2xk and S is a surface of the region bounded by the cylinder x+ y = 4…

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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: Use Green's Theorem to evaluate the line integral along the given positively oriented curve. ∫C…

A: 7/3 units