The surface r(u,v)=(u,2v,3u – 4v) is a (a) cone. (b) plane. (c) sphere.

Q: Find the plane tangent to the surface z=x x+ y° at the point (3,4,25). a. 6x+8y-z=25 O b. 6x-8y-z=25…

A: As we know that; The equation of the tangent plane: nxx-x0+nyy-y0+nzz-z0=0 And The normal vector at…

Q: The surface described by the conditions 2 + y? + z² = 4, x 0 is the same as the surface described…

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Q: Consider the surface z = (x+ 1)² + (y – 2)² – 1 Does the point (1, 3, 1) belong to this surface?

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Q: Explain briefly how Vø relates to surfaces of constant o(x, y, x). Hence, use Vø to find points in…

A: For the surface of constants ϕx,y,z=k, the gradient vector ∇ϕx0,y0,z0 is orthogonal to the level…

Q: A) || Jx2 + y2 dA , R is the disk x2 + y? < 9

A: We have to use polar coordinates here.

Q: F = (x² – y², y² – 2², z² – x²), S is the part of the plane r + y + z = 1 in the first octant.

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Q: Classify the surface: 1-z = 3y? + 4z? ,2

A: Given : The surface 1 - x = 3y2 + 4z2   To classify the given surface.

Q: וווכu nt that enclosed by the surface z = 12 - x 2 - y 2 and the plane z = A. A=1

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Q: Find the area of the surface. The part of the plane  z = 3 + 2x + 5y  that lies above the rectangle…

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Q: Evaluate (x2 + y? + z)dS, where S is the portion of the plane z = x + 2 that lies inside the…

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Q: a) Evaluate x²y? ds, where the curve C is the intersection between the surface z = := V1– y? and the…

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Q: Evaluate yds where S is the surface = x + 2y?, 0 <a < 1, 0<y S 2

A: Consider the surface, z = x + 2y²

Q: Determine the constants a, B, y such that the surfaces ax?z – xy? = B, yaz – y² = 0, - intersect…

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Q: Find the surface area of the plane ( 15x- 5z+20y=35) if it is in the first octant

A: Solution:-

Q: ? Find the surface area of the plane ( 15x-5z+20y%3D35) if it is in the first octant

A: Here we have to find the surface area of the plane 15x-5z+20y=35 ................... (1) for the…

Q: Find the area of the surface. The part of the sphere x2 + y2 + z2 = 16 that lies above the plane z =…

A: The objective is to find the area of the surface of the part of the sphere x2+y2+z2=16 that lies…

Q: 1. Find the area of the part of the surface z = a? + 2y that lies above the triangle with vertices…

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Q: Find the area of the part of the paraboloid z=x +y² that lies under the plane 2=9.

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Q: Sketch the surface z = 3x? + y? +1 and find its linear approximation at P-(0, -1)

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Q: Evaluate (image) over the sphere D = { (x,y,z) | x2 + y2+z2 ≤ R2 }

A: Given integral is ∭x2+y2+z2xyzdxdydz over the sphere D = { (x,y,z) | x2 + y2+z2 ≤ R2 } Change…

Q: The solid enclosed by the sphere x² +y² +z? = 4a* and the planes z = O and z = a.

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Q: Determine the intersection of the hyperbolic paraboloid z = (y2/b2) − (x2/a2) with the plane bx + ay…

A: Given The hyperbolic paraboloid equation is z=y2b2-x2a2 and the plane bx + ay − z = 0 .

Q: Find the area of the surface. The part of the paraboloid z = 1 – x² – y2 that lies above the plane z…

A: Givenz=1-x2-y2z=-2Suppose the surface is defined byz=f(x,y) for x∈R, then the area of the surface is…

Q: the trace of the surface in the plane x=1?

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Q: Find the surface area of the part of the plane z = 4+ 7x + 6y that lies above the rectangle [1, 6] ×…

A: We can use integral calculus to find the surface area of the part of the plane above the given…

Q: Check whether the given surface (x – a)² + (y – b)² + (z – c)? 16 is developable or not. -

A: Given surface(x-a)2+(y-b)2+(z-c)2=16orz=c+16-(x-a)2-(y-b)2If the surface is developable…

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: Find the area of the part of the sphere x2 + y? + z² = 4z that lies inside the paraboloid z = x2 +…

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Q: Consider the parametric surface S given by the portion of the sphere x +y² +z² = 4 where z 2 1.

A: In this question, we have to parameterize the surface in cartesian coordinates and then find the…

Q: Find the surface area of the part of the plane 7+ 2x + 5y that lies above the rectangle [1, 4] x [0,…

A: Given that  Z=7+2x+5y To find the area of the surface of the plane that lies above the rectangle

Q: Evaluate the surface z =1,0<x,y <1.

A: To solve this we will use divergence theorem.

Q: . $F•ds where F = (x, 2y, z+3x) and S is the closed surface (x– 1)? + y² = 4, 0<z< 3 including its…

A: F=x,2y,z+3x and S is the closed surface x-12+y2=4, 0&lt;z&lt;3 including its top and bottom dics.To…

Q: If a parametric surface given by r1(u, v) = f(u, v)i + g(u, v)j+ h(u,v)k and –1< u<1, –2 < v< 2, has…

A: We will find out the required value.

Q: Find the area of the surface cut from the paraboloid x2 + y2 - z = 0 by the plane z = 2.

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Q: Locate and classify the stationary points on the surface z=xy2-2xy-2x2-3x, remembering that  x, y ,…

A: Solving for stationary points of given function using partial derivatives

Q: Find a parametric representation for the surface. the part of the plane  z = x + 2  that lies inside…

A: We need to find parametric representation.

Q: The part of the surface z =1  + 3x+ 2y 2 that lies above the triangle with vertices (0, 0), (0, 1)…

A: Part of the surface: z=1+3x+2y2∂z∂x=3, ∂z∂y=4yVertices of the triangles are (0,0) , (0,1) , (2,1)

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: )الكــره the Sphere OThe Portion of planes z=l x +y+Z² = \6 between the %3D and z=2.

A: Volume of the sphere x2+y2+z2=16 or x2+y2=16-z2 is: V=∫z=12∬x2+y2≤16-z2dxdydz   =∫z=12π16-z2dz…

Q: a) Evaluate ds, where the curve C is the intersection between the surface z = V1- y? and the plane r…

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Q: Find the surface area of the part of the sphere x2+y2+z2=9 that lies above the cone z=√x2+y2

A: answer is below

Q: Sketch the surface z = f(x,y). f(x.y) = 4x2 + 4y2 + 2 O A. OB. OC. OD. N

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Q: Sketch the surface z = f(x,y). flx.y) = - +y? OA. O B. OC. OD. y

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Q: Find the extreme values of ƒ(x, y, z) = x2yz + 1 on the intersection of the plane z = 1 with the…

A: consider the following equations  f(x,y,z)=x2yz+1                                  (1)h(x,y,z)=z-1…

Q: Express the area of the given surface as an iterated double inegral in polar coordiantes, and then…

A: We need to find the surface area in the portion of the sphere x^2+y^2+z^2=16 between the planes z=1…

Q: The surface that is linear with the three variables, x, y, and z is called cylinder plane sphere O…

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Q: Find the surface area of the part of the plane z = 4 + 6x + 2y that lies above the rectangle [1, 4]…

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Q: (a) Find the surface area of the plane 3x +2y + z = 6 that lies in the first octant.

A: If z=fx, y then surface area is, AS=∫∫D1+zx2+zy2dA

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

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