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

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 surface area of the portion of the paraboloid given by the equation z = 1– x2 – y² that is…

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Q: Consider the unit sphere x² + y² + z² = 1. Find the average distance from its north pole to a point…

A: Given: x2+y2+z2=1 It is equation of sphere of radius r=1 unit. We have to find the average distance…

Q: (c) The cone r² = 4(y² + z?) for (0 <x < 4

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Q: Evaluate the surface integral. x dS, S is the part of the plane 16x + 8y + z = 16 that lies in the…

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Q: 1. 3 Consider the parametric surface 7(u, v) = (2uv, u? - v, u² +v²), where u +v < 2. Evaluate the…

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Q: state the type of the quadric surface and describe the trace obtained by intersecting with the given…

A: Given: x2+y42+z2=1  ……1   Equation of plane is: z=14  ……2

Q: Evaluate || f (x,y,z) dS where S is the portion of the sphere centered at the origin of radius two…

A: we have the equation of sphere x2+y2+z2=22s is the portion bounded between xy plane and z=1 in the…

Q: If A = 18zî – 12j + 3yk then find SS,A.ñds where S is the surface of the plane %3D 2x + 3y + 6z = 12…

A: Given, surface S: 2x+3y+6z=12           and                       A→=18zi^-12j^+3yk^

Q: In which part of the (x, y) plane the following equation is elliptic? Uzr + 4ury + (x² + 4y*) uyy =…

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Q: Which of the following surfaces could have contour map 5 2 1 1. cone 2. parabolic cylinder 3.…

A: Please see the below picture for detailed solution.

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

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Q: Determine the projection of the following surface on the xz plane. (xz trace) 4z – + = 0 O Parabola…

A: Solution is given below:

Q: Evaluate the surface integral: x dS,  S is the part of the plane  12x + 6y + z = 12  that lies in…

A: From the given statement, the plane is 12x + 6y + z = 12 and it is lies in the first octant.

Q: Find the centroid of the portion of the sphere x2 + y2 + z2 = a2 that lies in the first octant.

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Q: Consider the portion of the plane x + 2y + 3z = 12 that sits above the circle (x − 2)^2 + (y − 2)^2…

A: The solution are next step

Q: z+v" = 2 +3: = 2

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Q: state the type of the quadric surface and describe the trace obtained by intersecting with the given…

A: The given quadratic surface is  x2+y42+z2=1 , y=0

Q: 4. Consider the surface defined by the portion of the cone z = -Vx2 + y? that lies below the…

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Q: Find a parametrization for the line segment joining points (0, 2) and (4, 0) using the angle 0 in…

A: Modify the given figure as shown in Figure 1.

Q: Are there any points on the hyperboloid x 2 − y 2 − z 2 = 1 where the tangent plane is parallel to…

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Q: Please make sure its right. calc 3 Find a parametric representation for the part of the plane z = x…

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Q: Find all points on the isometric dot plane, Ax,1 yl, with x and y integers, that are ex- actly V133…

A: a)The distance from origin to point x,y is 133. b)The distance from origin to point x,y is 217.

Q: 3. Use polar coordinates to find the surface area of the paraboloid z = x² +y² – 5 that lies below…

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Q: Verify Stoke's theorem for A=(2x-y)i-yz²j-y*zk where S is the upper half 5A. surface of the sphere *…

A: Consider the given vector and sphere equation. A→=2x−yi−yz2j−y2zk and x2+y2+z2=1 Since S is the…

Q: Evaluate ⌠⌠(x+y)ds, where S is the cone whose parametric equations are x=ucosv, y=usinv, z=2u,…

A: Given  cone whose parametric equations are x=ucosv, y=usinv, z=2u, 0≤u≤3, 0≤v≤π/2 we find the given…

Q: Given the curve of intersection of the x² + 4y² -z = 0 which is a paraboloid and the plane 2x - z =…

A: Solution:-

Q: Consider the set S given by S = {(x, y, z) ER³ : 2 = 2x² - y²}, where A E R\{0} is constant.…

A: As per the question we are given a quadratic surface as : S = {(x,y,z) ∈ R3 | z =2x2 - λy2} And we…

Q: Find for the point P(x,y,z) on the plane 3x + y – z – 15 = 0 that is closest to the origin.

A: The distance of the point P(x,y,z) from the origin is: d2=x2+y2+z2 Now we have to minimize d if the…

Q: What is the equation of the plane to the surface z = e3Y sin(3x) +2 at the point P,0, 3) O x- y + ez…

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Q: Find all points on the surface xy - z2 = 1 which are closest to that origin. Let x,y,z be the…

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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: Sketch the curve of intersection of the surfaces. The ellipsoid 2x + 2y² + z? = = 24 and the…

A: We have to write the intersection curve

Q: Find an equation of the tangent plane (in the variables x, y and z) to the parametric surface…

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Q: Classify the quadric surface given by equation  and  determine its standard  form. x2 + y2 − z2 − 2x…

A: x2 + y2 − z2 − 2x − 4y − 4z + 1 = 0

Q: Please help.

A: Note: The tangent plane to the surface given by f(x, y, z) = 0 at (x0, y0, z0) has the equation,…

Q: Evaluate the surface integral fF- dS where F = (52,-52, 5y) and S is the part of the sphere a² +2+22…

A: Solution

Q: If a parametric surface given by ri (u, v) = f(u, v)i + g(u, v)j + h(u, v)k and -3 <u < 3, –4 < v<…

A: We solve this

Q: 1. Identify the parametric surface S given by: r(u, v) = 3 cos ui+ vj + 3 sinu k where 0 <u < 2n and…

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Q: The spherical surface S is tangent to the z=0 and x+y+3z−2=0 planes, with (−1,2,0) being one of the…

A: Explained below

Q: Find a set of parametric equalions for the surface formed when the curve y(1+z²) – 1 = 0 is revolved…

A: Find a set of parametric equations for the surface formed if the curve y1+z2-1=0 is revolved about…

Q: Plot the implicitly given surface (x − 1)² /4 + (y − 1)² +z² = 1 (a) using the sphere function. (b)…

A: As per the question we have to plot the following surface : (x-1)2/4 + (y-1)2 + z2 = 1 With the help…

Q: Find an equation of the surface of revolution by revolving the plane curve 3y = v(9- [(x-1)] ^2)…

A: Solution

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: Consider the planes 2x − 3y + z = 6 and −2x + 4y − 3z = 4 Find parametric equations x(t), y(t), z(t)…

A: Given :-  The given planes are : 2x -3y + z = 6-2x + 4y -3z = 4 To Determine :-  The Parametric…

Q: Parametrize the intersection of the surfaces using t = z as the parameter. (If there are multiple…

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

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