Evaluate || 4.nds , where A=zi+xj – 3y°zk and S is the surface of the cylinder x² + y =4 included in the first octant between z = 0 and z = 3.

Q: 2. Evaluate the triple integral //y o 2 = y, the xy-plane, and the parabolic cylinder y = 1- r2. y…

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Q: The area of the surface obtained by revolving the curve y = x3 for 1<x<8 about the y-axis, is given…

A: Surface area obtaining by x=f(y) from x=a to x=b is about y-axis given in step 2.

Q: 2. Evaluate the triple integral ///_y dV; G is the solid enclosed by the plane z = y, the xy-plane,…

A: To evaluate ∭GydV,  where G is the solid enclosed by the plane z=y, the x-y plane, and the parabolic…

Q: Evaluate the surface integral.     Sz2 dS, S is the part of the paraboloid  x = y2 + z2  given by…

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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 JS yz + 4ryds where S is the surface of the solid bounded by 4x+2y+z=84, z=0, y=0 and x=0.…

A: Use the divergence theorem to find the surface integral,…

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: Evaluate the surface integral. z + x2y x²y) ds S is the part of the cylinder y² + z2 = 9 that lies…

A: We have to find the surface integral from the given condition

Q: Evaluate the surface integral I F. dS when F(x, y, 2) = y²i+ 3x² j and S is the graph of - ža (1+æ²…

A: Here, Fx,y,z=y2i+3x2jz=121+x2-y2

Q: Evaluate (Vx A). ndS over the surface of intersection of the cylinders x + y a, = a which is…

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Q: 6. Evaluate the triple integral 2x2z +2yz dV where S is the solid that lies below paraboloid z = -r?…

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Q: The surface area of the surface generated by revolving y=1-x? for 0<x<1 about the x- axis is None s=…

A: surface area S =∫012πy.ds where ds=1+(dy/dx)2

Q: Q1 (9 M)/ Use Gauss's theorem to evaluate ff. F.ds where F= 4xi - 2y'j +z°k and S is the surface…

A: We have to evaluate the integral ∬SF·ds where F=4xi-2y2j+z2k and S is the surface bounding the…

Q: Evaluate the surface integral. y ds S is the part of the paraboloid y = x² + z² that lies inside the…

A: The projection of S onto the xz-plane is the disk D=x,z: x2+z2≤9. We have to calculate the surface…

Q: 1. Find the volume of the solid bounded by the paraboloids z = 6 – x² – y² and z = 2x² + 2y².Use…

A: From the given problem:   z=6-x2-y2  ; z=2x2+2y2   For finding volume we using the triple…

Q: Evaluate the surface integral. z2 dS, S is the part of the paraboloid  x = y2 + z2  given by  0 ≤…

A: given the surface integral  z2 dS  where S is the part of the paraboloid x= y2+z2   given by 0≤x≤1…

Q: Let E be the solid region in the first octant bounded by the coordinate planes, the cylinder…

A: The cylinder x2+y2=5y and the sphere x2+y2+z2=25. Obtain the triple integral: ∫∫∫Ey dV The solid is

Q: The area of the surface obtained by rotating the curve y = 1 I+ X, 1 <xse about the line x = 28 is…

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Q: 1. Find the volume of the solid bounded by the paraboloids z = 6- x² - y and z = 2x? + 2y2.Use polar…

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Q: Consider the solid Q that is generated in the first octant by the surfaces S1 : y+z = 4, S2 : y - 1…

A: Solution is below.

Q: 1. Evaluate the following surface integrals: (a (x + y) dS, o : part of the surface 2x + 3y + z = 6…

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Q: Evaluate SſF-dš where F = yi +2x j +(z-8)k and S is the surface of the solid bounded by 4x + 2y +z =…

A: By using the divergence theorem we solve the above question

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

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Q: 6. Below is the solid bounded by the surfaces S1:z° = 16 – 2y S2: x = 4-y + z, S3: x = 1, in the…

A: Given a solid described by,  s1:z2=16-2ys2:x=4-y+zs3:x=1.

Q: (a) Evaluate the following surface integral yex²+z² dS, where o is the portion of the surface of the…

A: Given the integral ∬σyex2+z2dS we have evaluate the given surface integral where σ is the portion of…

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: #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: 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: 1. Evaluate the following surface integrals: || Vx2 + y2 + z² dS, o : the part of the surface of the…

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Q: Evaluate the the triple integral ||| y dV; G is the solid enclosed z = y, the xy-plane, and the…

A: Evaluate triple integral:- ∫∫∫y dv ∫∫∫y dv=∫01∫01-x2∫0y ydz dy dx ∫∫∫y dv=∫01∫01-x2 yz0y dy dx ∫∫∫y…

Q: (b) Evaluate the surface integral S.S x² dS over the surface x² + y² = 1 between z = 1 and z = 4.

A: x=cos(u), y=sin(u), z=v  0≤u≤2π, 1≤v≤4 S(u,v)=&lt; cos(u), sin(u), v &gt;

Q: 15. Evaluate SLL:(z+1)dV where E is the solid enclosed by the paraboloid y = x2 + z² and the plane y…

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Q: Find the area of the surface generated by e +e -y eY + e -y in the X= revolving the curve x= -In 3…

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Q: Find the surface area of the surface resulting from revolving the cu y=x+2 on the interval [0,1]…

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Q: Here is a problem where it is challenging to even figure out what the surface looks like. Consider…

A: The solution is in step 2 with clear information and step by step wise.

Q: 1. Evaluate the integral: SSS, x² + y² + z² dV where D lies above the cone z = spheres x? + y? + z²…

A: Here we have to evaluate the integral value, where D lies above the cone z= √(x2+y2) and between the…

Q: x² and t %3D 2) = z where E is the region bounded by the parabolic cylinder y

A: We have to set up the integral according to the given information

Q: 2) Find the area of the surface r(u,v)=u cos vi+u sin vj+2vk where 0su<3 and 0<vs 2n .

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

A: Given that The surface integral is ∬Sxy dS, where S is the quarter cylinder x2+y2=1; x,y≥0 between…

Q: The surface area of the surface generated by revolving y 1-x? for 0Sr<1 about the x-axis is S=…

A: Rule- If the curve y=f(x), a≤x≤b is rotated about the x−axis, then the surface area is given by

Q: Which of the following represents the are a of the surface of the portion of the cylinde 2+ z? = 4…

A: We need to find the integral which represents the area of the given surface. This can be done as…

Q: Evaluate SF. NdS where F(x, y, z) = xi + yj + zk__ and the surface S is given by S S: z=4-x² - y²…

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Q: 1. Evaluate the surface integral [[' xz² ds ; o is the part of the cone z = x² + y² that lies…

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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: a) Evaluate the surface integral ſ 2x²y dS over the surface y² + z² = 1 betwee x = -1 and x = 5.

A: a We have to evaluate the surface integral ∬σ2x2ydSover the surface y2+z2=1 between x=-1 and x=5.…

Q: S Find the voleime of the solid generated by revolving the region baunded by the given lines Cnd…

A: Hint: Try Disk Method to find the volume.

Q: 1. Evaluate the following surface integrals: (a) (x + y) dS, 0 : part of the surface 2x + 3y + z = 6…

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Q: Let S be the solid obtained by rotating the region shown in the figure below about the y-axis. y…

A: The function of the curve is y = 6x(x-1)2.

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: 2. Evaluate the triple integral / || y dV; G is the solid enclosed by the plane z = y, the xy-plane,…

A: Solution is given below;