a) Evaluate f, (2x – z) dS, where S is part of the plane 3x + 2y + z = 1 that lies above the rectangle 0 < x < 2, 1< y<2.

Q: z dv, where E is enclosed by the paraboloid z = x2 + y² and the plane z = 4. Evaluate

A: see 2nd step

Q: Evaluate ff.dS, where (a) v = (xy, -x², x +z) and S is the part of the plane 2x + 2y + z = 6…

A: Problem is solved using concept of double integral.

Q: B- Consider the following subsets of real plane R² Determine with graph: 1- Arcwise connected A =…

A:

Q: Evaluate ff, F.dS, where F= (4xz, -y², yz) and Sis the surface of the cube bounded by the planes x =…

A: We will use divergence theorem to find given integral  ∫∫SF→dS →= ∫∫∫EdivF→ dV Where E is  solid…

Q: 2 Consider the line L: y -1 +t|0, te R, and the plane T : x – 4y – z = 6. (a) Show that L is…

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Q: parametrizes the portion of the sphere x² + y² + z² = 9 that is above the xy-plane and below the…

A:

Q: 25. Evaluate F.N dS where F= yl + 2xj –- zk and S is the surface of the plane 2x +ye 6 in the first…

A:

Q: 4. Consider the curve in the plane defined by r3+y3 = 6xy. %3D (a) Find for any point (r, y) on the…

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Q: C is the hexagon in the xy-plane with vertices (5, −5), (5, 5), (0, 10), (−5, 5), (−5, −5), and (0,…

A: Given : Vertices : (5,-5) , (5,5), (0,10), (-5,5), (-5,-5) and (0,-10) Integral given :…

Q: Show that the area of that portion of the surface of the paraboloid x² + y² = 4z included between…

A: Here we need to find area between given plane and x*x + y*y = 4z

Q: Let f(x, y, z) = x² + 2y? + 4z2. Find the point on the plane x + y + z = 14 at which f has its…

A: Given: fx,y,z=x2+2y2+4z2and the plane equation x+y+z=14 To find the point x,y,z The Lagrange's…

Q: a) Evaluate (-ri-yj+2z k) n dS, where a is the portion of the paraboloid : 3x +3y between the planes…

A: We have to Evaluate the Integral using polar coordinate

Q: show. that theAine in.which.the planes.X+.2y-22=-5 nod5X-2y.z=.0...jntersec.is..pavalle.to.theine.…

A:

Q: a) Evaluate ff, (2x – z) dS, where S is part of the plane 3x + 2y +z = 1 that lies above the…

A:

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: Enclosed by the paraboloid z = x² + 3y? and the planes x = 0, y = 1, y = x, z = 0

A: We know that, volume under surface z=fx,y and above the region D in xy-plane is given as…

Q: What is the projection of r(t) = ti + t4j + etk onto the xz-plane?

A: Given: r(t)= ti+t4j+etk

Q: I, Use Stoke's theorem to evalueate S.Fidr, where F Cau2)= - it +x3 + 2î aind C is the Ceurve of…

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Q: Let R = (c, 1,-1) where c denotes a real number. Let ä denote the plane x + y – z = 2. Find the…

A: Given Information -  The distance from the point R = (c,1,-1) to the plane x+y-z = 2 is 2. We have…

Q: ind the equ.of Plane that Contains plos1,3) and Parallel to the X-2 Plane and PerPendicular to the…

A: Solution:-

Q: Evaluate x* zdS, where S is the part of the plane z = 1+4x + 3y above the rectangle [0, 3] x [0, 5]

A: Please see the below picture for detailed solution.

Q: 2. Do the three planes x + 2y + z = 4, y - z = 1, x+ 3y= 0 have at least one commom point of…

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Q: Find the projection in the ry plane of the curve of intersection of the surfaces 222 = r² + y? and r…

A: Find your answer below

Q: Describe the intersection of the horizontal plane z = h and thehyperboloid −x 2− 4y 2 + 4z 2 = 1.…

A: Put z=h in given equation -x2-4y2+4h2=1⇒x2+4y2-4h2=-1

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: Prove that the distance d from a point P (xo, Yo, 2o) to the plane r: Ar + By + Cz + D = 0 is given…

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Q: Зх + 2у + 3z %3D 6

A: Use a graphing utility to plot the graph of 3x+2y+3z=6 .

Q: Зу — 2z 3D 6

A: Use a graphing utility to plot the plane, 3y-2z=6 .

Q: Show that the minimum distance from the origin to a point on the plane ax + by + cz = d is |d| Ja? +…

A: Given :               ax + by + cz = d         We know that minimum distance (p) of a point x1,y1,z1…

Q: Find the distance from a point P(x1, y1, Z1) to the plane Ax + By + Cz = D.

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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…

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Q: Evaluate x2 zdS, where S is the part of the S plane z = 1+ 2x + 3y above the rectangle [0, 2] × [0,…

A: We have to evaluate ∬Sx2zdS, where S is the part of the plane z=1+2x+3y above the rectangle 0,2×0,3.…

Q: 2. Do the three planes X + 2y + z = 4, y - z = 1, x+ 3y = 0 have at least one commom point of…

A: The equation of the plane is given by the formula ax+by+cz+d=0. The system of planes can be solved…

Q: Calculate F- dS if F(x, y, z) = x²ye²i+ æy?j – 2xye*k, where S is the surface of the box bounded by…

A:

Q: Consider the following subsets of real plane R² A = {(0,y): ≤ y ≤ 1} B = {(x, y): 0 ≤ x ≤ 1, y =…

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Q: Find the projection in the ry plane of the curve of intersection of the surfaces 2z2 = r? + y? and…

A: Let's find.

Q: In space coordinates, the xy plane consists of all the points of the form O (0, y, z) O (X, y, z) O…

A:

Q: a) Evaluate ff, (2x – z) dS, where S is part of the plane 3x + 2y + z = 1 that lies above the…

A: Surface integral

Q: The part of the plane 3x + 2y +: - 6 that lies in the first octant

A:

Q: a) Show that the line I = 0, y = t, z =t where te R a) lies in the plane 6z+4y–4x = 0 b) is parallel…

A: In the question it is asked to solve for the planes having equation given below.

Q: 3. Show that every sphere through the circle x2 +y-2ax+r²=0, z=0 cuts orthogonally every sphere…

A: This is a co-ordinate geometry problem

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: Show that a indeed perojective plane of gives 0rder BİBD (n+n+,n+),!

A: We need to prove that a projective plane of order n indeed gives a BIBD (n2+n+1, n+1, 1)

Q: Evaluate JJ (² where o is the first-octant portion of the plane 2x + y + 2z = 6 (x² + 2yz)dS

A:

Q: Find the point lying on the intersection of the plane x + + z = 0 and the sphere x2 + y2 + z2 = 25…

A: According to the given information, it is required to find a point lying on the intersection of the…

Q: a) Evaluate //(-ri-yj+2: k) n dS, where a is the portion of the paraboloid : = 3r - 3y between the…

A:

Q: Find the extreme values of f(x,y,z) = x-yz + 1 on the intersection of the plane z= 3 with the sphere…

A:

Q: 7. Find the area of the portion of the plane z = 2y that lies directly above the circle x +(y-1)² =1…

A: we have to find area of the portion of plane .and coverting the problem in polar coordinate makes it…