Find f[fp Vx2 + y2 + z² dV, where D is the sphere of radius a centered at the origin.

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: 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: 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: Let B be the ball with center at the origin and radius 1 SS $3(x² + y? + z²)dV = B

A: Solve integral by using Cylindrical cordinates

Q: Find the point on the surface z=x2-y2 at which the tangent plane is parallel to the plane…

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Q: 28. F(x, y, z) = x³i+y°j+z°k, and S is the sphere of radius 3 and centered at the origin.

A: divF→=ddxx3+ddyy3+ddzz3        =3x2+3y2+3z2        =3x2+y2+z2  by divergence theorem,…

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: Evaluate e dV where E is enclosed by the paraboloid z = 5+x +y, the cylinder a2 + y = 1, and the xy…

A: The objective is to evaluate ∫∫∫EezdV, where E is enclosed by the paraboloid z=5+x2+y2, the cylinder…

Q: Find a parametrization of the paraboloid given by f (x, y) = x 2 + y2.

A: To find a parameterization of the paraboloid given by: fx,y=x2+y2

Q: Show that the value of ∮C xy2 dx + (x2y + 2x) dy around any square depends only on the area of the…

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Q: Find the orthogonal trajectories on the sphere x2 + y2 + z2 = a2 of its intersection with the…

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Q: Let B be the ball with center at the origin and radius 1 SS [3(x² + y² + z²)dV =

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Q: b) Use Gauss' theorem to evaluate f[ F· ndS where M UTM = (x² – y)i + y°z²j + (xy – y²z³)k and o is…

A: The given problem is to evaluate the given double integral using gauss divergence theorem over the…

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: Consider the PDE Y Urr + x Uyy + Ux – Uy x² + y? In what region of the xy- plane is this PDE…

A: compare the given equation with Auxx+Buxy+Cuyy+Dux+Euy+Fu=G A=y,B=0,C=x,D=1,E=-1 calculate B2-4AC…

Q: Use the method of Lagrange multipliers to find the absolute extrema of the f(x, y, z) = y + xz² x² +…

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Q: Evaluate /|/. x dV where E is the solid enclosed by the paraboloid x = y²+z² and the plane x = 9

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Q: Let B be the ball with center at the origin and radius 1 SSS 4(z² + y² + z²)dV = B

A: Explanation of the answer is as follows

Q: (a) 'Find the orthogonal trajectories on the surface of the sphere x' + y? + z = 0 with the…

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Q: Let B be the ball with center at the origin and radius 1 3(x² + y + z?)dV = %3D B

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Q: Evaluate z² dv. where E is the region above the zy-plane below the cone z = 2+2 and inside the…

A: The given problem is to evaluate the given triple integral over the given region E bounded above by…

Q: Find all points on the surface at which the tangent plane is horizontal. (a) z = x³y² (b) z = x² −xy…

A: (a). Given the surface z=x3y2,the tangent lane at a given point is given by :…

Q: Use the Divergence Theorem to evaluate (6x + 7y + z2) dS where S is the sphere x2 + y2 + z? = 1.

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Q: Evaluate JJJ, (x² + y²)dV where D is the region that lies above the cone z = c/x² + y2 and inside…

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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: FD is the region in the first octant between the spheres x2+y2+z²=1 and x2+y2+z²=16, then dV =

A: Converting into spherical coordinates

Q: Evaluate the triple integral (* + 1y) dV where E is bounded by the parabolic cylinder y = 32² and…

A: We have to find the triple integral  ∫∫∫E(x+y)dVwhere E is bounded by the parabolic cylinder y=3x2…

Q: Consider D to be the region below the paraboloid z = 3+x² + y² , inside the cylinder x + y² =1 and…

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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: Evaluate || F.ndS F = xi + yj+(z² –1)k if and o is the curved surface of the cylinder y = \4-x? from…

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Q: Q7. Use Green's theorem to evaluate fy² dx + 3xy dy, where c is the boundary of the semi annular…

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Q: Find the absolute extrema for f(x, y) = (x² + 2y²)e-x²-y' over the closed disk D: x° + y° < 4. 2.5 2…

A: Given a function of x and y. We need to find the absolute extrema ( maxima and minima) over a given…

Q: Let B be the ball with center at the origin and radius 1 SS f6(n² + y² + z² )dV

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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 maximum value of f(x,y,z)=25x+3y+17z on the sphere x2+y2+z²=100

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Q: Find the average height of the hemispherical surface z = √(a2 - x2 - y2) above the disk x2 + y2 ≤ a2…

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Q: Find the sum of maximum and minimum values of the function f(x, y, z) = 8x + 16y + 24z on the elipse…

A: To determine the maxima and minima of the function f(x, y, z) with respect to the function is…

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: Calculate x dy dz + y dz dx + z dx dy where is formed by the upper part of the sphere x² + y? + z2 =…

A: To calculate            ∯σ x dy dz+y dz dx+z dx dy where σ is formed by the upper part of the sphere…

Q: find the flux of f=zk across the portion of the sphere x2+y2+z2=a2 in the first octant in the…

A: Given the field, F=zk The surface S is the portion of the sphere x2+y2+z2=a2 in the first octant in…

Q: Find the maximum value of f(x,y,z)=9x+14y+29z on the sphere x2+y2+z²=25

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Q: Find the flux of F = (y +xz)i + (y + yz)j – (2x + z²)k upward through the first octant part of the…

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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: ind the average height of the (single) cone z = sqrrt(x2 + y2) above the disk x2 + y2&lt;= a2 in the…

A: Given equation of cone is, Bounded by the disk,

Q: Find the area of the part of the paraboloid x = y2 + z2 that lies inside the cylinder y2 + z2 = 1

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Q: calculate ∫ ∫ ∫Ty dv, being the T region inside the paraboloid z= 1/(x2+y2)the sphere x2+y2+z2=5

A: We need to calculate the triple integral ∫ ∫ ∫Ty dv, where T being the region inside the paraboloid…

Q: Find the points on the surface z²2=xy+1 that are closest to the origin using LaGrange multipliers

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Q: Find a parametrization for the intersection of the surfaces: z = x2 + y2 and  x2 + 4y2 = 1

A: Find the parametric representation for the intersection of the given surfaces.

Q: Evaluate the surface integral. v² as S is the part of the sphere x2 + y2 + z2 = 64 that lies inside…

A: A scalar function f(x,y,z) is a rule that is used to associate a number or quantity with a…