18. Evaluate J (V x F) · N dS where F = yi + (x- 2xz) j- xy k and S is the surface of the sphere x +y +z = a², above the xy-plane.

Q: 7. D is bounded by the surface (r2 + y² + z²)² = a²(x² + y² – 22). %3D

A: We have to find volume where D is bounded by the surface (x² + y² + z²)² = a²(x² + y² — z²).

Q: Find the maximum and minimum values of ƒ(x, y, z) = x - 2y + 5z on the sphere x2 + y2 + z2 = 30.

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Q: 3.4.8 Evaluate the surface integral ff, U(x, y, z) dS, where S is the surface of the paraboloid z =…

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Q: Find the maximum value of f(x,y,z)=2x+5y+13z on the sphere x²+y²+z²=36

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Q: Evaluate the surface SS₂ xz ds S is the part of the 2x+2y+z=4 that li- the first octant.

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Q: 1. Let F(x, y, z) = yi+ xj+ yzk and the surface S be the part of the paraboloid z = 4 – x2 – y? that…

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Q: 1. Let S C R be the surface defined for all 0 < x < 1, -1 < y < 1 by z = r² + y. Compute ffsa ds

A: It can be solved by partial differentiation and integration

Q: (a) Prove that M: (x² + y^y° + 3z² = 1 is a surface. (b) For which values of e is M: z(z – 2) + xy =…

A: Consider the provided question, (a)

Q: Evaluate yze-y dS V(1+ x²)e2v + 1 where S is the surface z = xe", 0 < x < 2, 0 < y < 3.

A: Given: ∬Syze-y1+x2e2y+1dS            where Sis the surface z=xey, 0≤x≤2, 0≤y≤3.

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: 16. Let F(x. y, z) = xyi+ xzj+yzk and C be the curve of intersection between the sphere z² +7+z² =…

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Q: 18. Let f(x, y, z) = 2x + 4y – 4z. Find the minimum and maximum values on the sphere x2 + y² + z² =…

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Q: the surface z = &x + 4y th=

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Q: B Suppose we want to find all the points on the surface gx, y) x+ 4y - 4 where the tangent plane is…

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Q: IF F= 3yI - Xy I+ 9ZK and s is the Surface of the Pasabolied 2X =x2 fy Bounded By Evaluate %3D I=2…

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Q: 17. Find the exact area of the surface z = 1 + 2x + 3y + 4y2, 1x 4, 0 y < 1

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Q: The attached figure shows the surface S1: y + z= 2, limited in the first octant with surface S2: x =…

A: The area of a surface z=f(x,y) bounded by the region R is given by the integral expression:-…

Q: 3. Evaluate , (x + z)dS where the surface S is the first-octant portion of the cylinder y² + z² = 9…

A: Step:- 1 Since, we know Surface integrals ∫∫Sf(x,y,z)dS=∫∫Df(x,θ) rx×rθ dxdθ…

Q: If F = 2y i - zj + x²k and S is the surface of the parabolic cylinder y²= &x in the first octant…

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Q: sketch the given surface. x2 − 4y2 = z

A: Given: x2-4y2=z Divide the equation both sides by 4. x24-4y24=z4x222-y21=z4 The equation…

Q: 2. Evaluate the triple in tegral where E is the y-2x2+2z20nd the plane y= 8. solid bounded by SI…

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Q: 2. Find the area of the part of the sphere x² + y² + z² = 4z that lies inside the paraboloid z = x²…

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Q: 2. Evaluate f, z³ /x² + y² + z² dV where E is the solid hemisphere that lies above the xy-plane and…

A: Solution:- equation of sphere having centre origin (0,0) and radius=1 x2+y2+z2=1…

Q: 18. Evaluate § (9 − x² − y²) dV, where H is the solid hemi- sphere x² + y² + z² ≤ 9, z ≥ 0.

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Q: Find the points on the surface z2 = xy + 4 closest to the origin.

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Q: Find the points on the sphere x +y +z = 154 where f(x,y,z) = 3x + 8y + 9z has its maximum and…

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Q: Verify Stoke 's Theorem for F = (x² +y – 4) î + 3 xyj + (2 xz + z²) k over the surface of hemisphere…

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Q: 1. The part of the sphere x? + y² + z² = a² that lies within the cylinder x² + y² : = ax and above…

A: To Determine :-  The part of the sphere x2 + y2 + z2 = a2 that lies within the cylinder x2 + y2 = ax…

Q: Evaluate V x F · N dS using Stoke's theorem, where F = (2x – y)I- yz }-ýzk and S is the upper half…

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Q: 2. Evaluate ([.(Vx F). ds, where S is the surface of the paraboloid x2 + y2 + z = 4 above the…

A: We have given F→=(x2+y-4)i+3xyj+(2xz+z2)ksurface is given x2+y2+z=4 and above the xy planediagram of…

Q: a) Evaluate x²z dS where o is part of a sphere x² +y² + z² = 8 above %3D the plane 2.

A: Given that, ∬sx2zds And, x2+y2+z2=s⇒z2=s-x2-y2 Take the derivative,…

Q: (a) Compute the area of the portion of the cone x2 + y2 = z? with z 2 0 that is inside the sphere x2…

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Q: 3.4.8 Show that the sphere x² + y² + z² = 1, z ≥ 0. JJ,z²dS=, where S is the surface of

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Q: 10/ Compute e Fids, where S is the surface z =N16-x²-y² oriented up ward and F = <y,-x, zZ %3D 工=?

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Q: JJs where S is the sphere x² + y2 + z2 = 1.

A: Given S is the sphere                         x2+y2+z2=1 To evaluate:             ∫∫S3x+3y+z2dS…

Q: 4. Evaluate //F FdS, where F(x, y, z) = (2x + 2xz, -4xy², y - z2) and R R surface of the solid…

A: The given problem is to evaluate the given surface integral over the given solid surface bounded by…

Q: Evaluate ∬S(x2+y2)dS, where S is the surface z=4−x, 0≤x≤2, 0≤y≤2.

A: Given z=4-x , 0≤x≤2 ,0≤y≤2 to evaluate ∬Sx2+y2dS where S is surface

Q: Evaluate /F. .7 dS where F = (³+y²z)7+(y²+a²y)7+(2x²z+y^) k" and S is the surface of the cylinder x2…

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Q: XII. $F•d$ where F = (x,y,z²) and S is sphere of radius 2 with center at 0.

A: F→=x,y,z2, surface S is sphere of radius 2 with centre O=0,0,0To find: ∯SF→.dS→

Q: Find the points on the surface xy + yz + zx - x- z =0 where the ant pla

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Q: Q– Fin the surface area of the portion of the paraboloid z= 6-2x - y that is above the x-y plane…

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Q: Find the maximum value of f(x,y,z)=9x+15y+9z on the sphere x2+y2+z²=100

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Q: Let C be the intersection of the surface x² + y² + z² cylinder x² + 3y² = 4, z > 0. = 9 and the

A: As per the question we are given the equation of a sphere and a cylinder (with elliptical cross…

Q: E is the region behind the surface y = 4 – x² that is in front of the region in the xy-plane bounded…

A: Given integral: ∫∬E18x dV Given that E is the region behind the surface y=4-x2 that is in front of…

Q: 7. D is bounded by the surface (a2 + y² + z²)² = a²(x² + y? – z²). %3D

A: Sol

Q: (a) Given that F(x,y,z) = (3x² + 3y²)i – 2xyj + yzk, evaluate S, F·dr for which C is the boundary of…

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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: 15 - Consider the surface defined by z = f(z, y) = x³ + 2x²y +y². Find the equation of a normal line…

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Q: Are there any points on the hyperboloid r² – y² – 2² =1 where the tangent plane is parallel to the…

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Q: Find the points on the surface xy – z2 = 1 that are closed to the origin.

A: Given surface is xy-z2=1. let xy-z2-1=0⇒z2-xy+1=0 Now distance from origin is…