Evaluate [[[(x² + y² +=?)2 dV, where 2 is the solid hemisphere (half of the Ω sphere) enclosed by x + y +z = 1 and above the plane z = 0. а) 0 b) 2л d) 3 e) л

Q: Evaluate the surface integral F. dS where F = (4x, -5z, 5y) and S is the part of the sphere x² + y +…

A: The solution is given below:

Q: Find the center C and the radius a for the spheres in Exercises 51–52. 51. (x + 2)2 + y2 + (z - 2)2…

A: To determine the centre and radius of the sphere.

Q: Is the set c~(R* U {0}) “the entire plane without the non-negative reals" conformally equivalent to…

A: We need to evaluate the statement "the entire plane without non negative reals, ℂ-{ℝ∪{0}}, is…

Q: y dV, where B is the portion of the unit ball B = {(x, y, z) : x² +y² + z² < 1} which lies below the…

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Q: Let D be the solid in space lying beneath the ry-plane and between the sphere r2 + y? + z² = 3 and…

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Q: Evaluate f, A.n ds, where A = zi +j- 3y²zk and S is the surface of cylinder x? + y? = 16 included in…

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Q: Use Stoke’s Theorem to evaluate the line integral ∮C(y+2z)dx +(x+2z)dy +(x+2y)dz, where C is the…

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Q: (b) Let the distance between two adjacent planes parallel to the plane (hkf) in the direct lattice…

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Q: Evaluate e dV where E is enclosed by the paraboloid z = 4+ x² + y?, the cylinder E x2 + y? = 1, and…

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Q: Evaluate [f[(x² +y² +=?)2 dV, where 2 is the solid hemispi sphere) enclosed by x +y +z =1 and above…

A: We have to choose the correct answer from the given options

Q: Consider the area of the part of the plane z=ax+by+c that projects onto some region D in the…

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

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Q: a) Evaluate (-ri-yj+2z k) n dS, where a is the portion of the paraboloid : 3x +3y between the planes…

A: We have to Evaluate the Integral using polar coordinate

Q: Evaluate the surface integral. ∫∫sxdS, S is the part of the plane 6x+3y+z=6 that lies in the first…

A: The plane has intercepts at (1,0,0), (0,2,0), (0,0,6). The parameterize the surface S by. where

Q: 6. Use Stoke's Theorem to evaluate fF dr, where F =, and C is the part of the paraboloid z = 4 - x2-…

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Q: Exercise 11. Compute the area of the pseudo-sphere, i.e. the surface of revolution obtained by…

A: Area= ∫2πrds=2π∫r.dr/sin(ϕ)∫2πrds=2π∫r.dr/sin(ϕ) Due to tangent property: r/sin(ϕ)=a,r/sin(ϕ)=a, and…

Q: Find the surface area of that part of the plane 9x + 6y + z = 4 that lies inside the elliptic…

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Q: Use the Divergence Theorem to evaluate (5x + 10y + z) ds where S is the sphere x2 + y2 + z2 = 1.

A: In the given problem, S is the sphere x2+y2+z2=1Divergence Theorem:∬SF·n dS=∭Ediv F dVNow, the…

Q: find volume 26. The wedge cut from the cylinder x2 + y? = 1 by the planes z = -y and z = 0

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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: Given: The surface integral, ∫∫Sx2yz dS, S is the part of the plane z=1+2x+3y that lies above the…

Q: 8. Find the surface area of the part of the sphere x2 + y? + 22 : = 4z that lies inside the…

A: The solution is given as

Q: Let F(z, y, 2) = 12?ri + (ty + tan(z))j + (1z?z+ 4y?)k. Use the Divergence Theorem to evaluate f, F…

A: We must apply the Divergence theorem on a closed solid, but the surface S is not closed (it is just…

Q: Show thata perojective plane of gives 0nder indeed BİBD (+n+l, n+),1)

A: We need to show that the projective plane of order n indeed gives a BIBD (n2+n+1,n+1,1)

Q: Let G be the solid in the first octant bounded by the sphere 2x? + 2y? + 2z? = 32 and the coordinate…

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Q: a) Evaluate [-ri-yj+2: k) - n ds, (-a i- where a is the portion of the paraboloid z = 3r + 3y…

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Q: Let E be the part of the unit sphere r² + y²- = 1 where z > vx² + y². Calculate 22 dA.

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Q: Let F = . Use Stokes' Theorem to evaluate curlF · d§ , where S is the part of the paraboloid z…

A: Given: F=-yz,xz,xy To Find: Use Stokes' Theorem to evaluate ∬Scurl F·dS, whereS is the part of the…

Q: Evaluate e dV where E is enclosed by the paraboloid z = 5 + x² + y°, the cylinder 2² + y? 1, and the…

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Q: Consider the area of the part of the plane z = ax + by+C that projects onto some region D in the…

A: If the area of the region D on xy-plane is k. That is, ∫∫DdA=k. If the area of the some region of…

Q: Evaluate J CurIF. [cariF- nas F(x,v,z) = - yi + xj + zk, and S is part of the sphere z =9-x2 - y?…

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Q: Evaluate fff(x +y +:' dV, where a is the solid hemisphere (half of the sphere) enclosed by x +y+ =1…

A: Here in the given question, find the integration of the given function:∭Dx2+y2+z232dVWhere Ω is the…

Q: Use Stoke's Theorem to evaluate the line integral / 7ydx + 3zdy + dz where C is the intersection of…

A: Use Stoke's Theorem to evaluate the line integral ∫C7y dx+3z dy+ dz, where C is the intersection of…

Q: (b) Evaluate SSI, zdV, where D is the solid tetrahedron bounded by the four planes x = 0, y = 0,z =…

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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: Evaluate e dV where E is enclosed by the paraboloid z = 4+x² + y°, the cylinder x² + y? = 3, and the…

A: Here we have to evaluate ∫∫∫E ez dV , where E is the enclosed by the paraboloid z=4+x2+y2 , the…

Q: Use the Divergence Theorem to evaluate (4x + 7y + z²) dS where S is the sphere x2 + y2 + z² = 1.

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Q: Give an example of a parametrisation of the unit circle, C = {(x,y) E R² |x²+y² = 1}, that generates…

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Q: Given: ABCDE and FCG, C is the midpoint of BD Z1 = 2 4. Prove: ZF = ZG F/ A1 В C DE G Describe a…

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Q: b) Use Gauss' theorem to evaluate SS F · ndS where F = (x² – y)i + y°z²j+ (xy – yz³)k and o is the…

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Q: . Let F(x, y, z) = (x³, y³, z³). Use Gauss's theorem to find ffsF dS if S is the sphere x? + y? + 22…

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Q: Evaluate (xr + y² + =³)? dV, where 2 is the solid hemisphere (half of the Ω sphere) enclosed by x +y…

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Q: se the Divergence Theorem to evaluate (8x + 12y + z2) ds Is here S is the sphere x2 + y2 + z² = 1.

A: To compute the below integral using divergence theorem.

Q: Evaluate T// y dV, where E is the portion of the sphere x² + y? + z² = 9 in the first octant (where…

A: Let, x=rsinθcosϕy=rsinθsinϕz=rcosθ

Q: Evaluate e dV where E is enclosed by the paraboloid z = 5 + x? + y?, the cylinder x? + y? 5, and the…

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Q: Let F(x, y, z) = (x,y,z). Compute ∫∫sF · dS where S is the part of the plane x + y + z = 1 in the…

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Q: a) Evaluate //(-ri-yj+2: k) n dS, where a is the portion of the paraboloid : = 3r - 3y between the…

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Q: Let F = · Use Stokes' Theorem to evaluate curi curlF · d§, where S is the part of the paraboloid z…

A: here i had written stokes theorem and done some basic calculation to find the curve of intersection…

Q: Evaluate z dv, where E is enclosed by the paraboloid z = x² + y2 and the plane z = 16.

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Q: Evaluate F. dS, where F = 6xy²i + 6x²yj + 2z°k and S is the surface of the unit sphere.

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Q: Evaluate the following double integral dydr, -here D is the entire plane

A: The question is taken from the calculus in which we have to find the double integral of given…