S y = 3 – a 4z – 4 = 3(x + 1)²

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: Use Green's theorem to find the surface integral of the following geometry if V = xy'i - x³yi, and…

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Q: Consider the rormal line to the surface z= z + y? (parabokoid) at the point P(1,1, 2) Find the point…

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Q: Evaluate the surface integral. || x²yz ds, S is the part of the plane z = 1 + 2x + 3y that lies…

A: Given: ∫∫sx2yzdS where S is the part of the plane z=1+2x+3y that lies above the rectangle…

Q: Find the exact area of the surface  z = 1 + 2x + 3y + 4y2, 1 ≤ x ≤ 9, 0 ≤ y ≤ 1.

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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: 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: Evaluate the surface integral. xz ds, S is the part of the plane 2x + 2y + z = 4 that lies in the…

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Q: 11. Find the area of the surface obtained by rotating the curve y = a, 0 < a < 3/4, about the a…

A: Surface area of revolution

Q: Use Stoke's theorem to evaluate the integral xydx +2x 'dy-&r 'zdz where C is the boundary of the…

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Q: Let S be the part of the hyperbolic paraboloid z = x2- y2 located between the cylinders x2 + y? = 1…

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Q: Let S be the portion of the surface x -2x + y² + z² = 8 for 1<z<2. Find the area of the surface S.

A: x2-2x +y2+z2 =8 ⇒(x-1)2+(y)2+(z)2=(3)2 So the surface S is a circular patch of a circle canter at…

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: find the surface area of the portion of the surface z=4-x2/2 that lies above the region bounded by…

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Q: 15- Evaluate the surface integral f, xy dswhere x is the part of the plaue y = 1-x-z that lies in…

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Q: Find the exact area of the surface z = 1 + 2x + 3y + 4y2, 1 s x< 11, 0 < y 1.

A: Consider the given: z=1+2x+3y+4y2 So, area of the part of the surface: Find the first order of the…

Q: Evaluate the surface integral  ffR(x+y)dS where σ is the portion of the plane z = 6 − 2x − 3y in the…

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Q: Evaluate the surface integral. /| xz ds, S is the part of the plane 2x + 2y + z= 4 that lies in the…

A: We need to find surface integral.

Q: Use Stoke's theorem to evaluate the integral xydx +x*dy– 4x'ydz where C C is the boundary of the…

A: Given: ∫Cxydx+x2dy-4x3ydzBoundry square surfaces z=1vertices are (1,0,1),(1,1,1) &amp; (0,1,1) &amp;…

Q: Q2 (a) Find the volume under the surface z = 3x³ + 3x?y and over the rectangle R = {(x, y):1 < x <…

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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: (b) Verify Stokes' theorem on a region which is a portion of the sphere x? +y? +z² = 16 which lies…

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Q: Given a surface 2xy - In (3z- 2) = 10, wherez=f(x, y). Find zx at y =7, z= 1. Leave your answer…

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Q: Evaluate the surface integral. || x²yz dS, S is the part of the plane z = 1 + 2x + 3y that lies…

A: Given: The surface integral, ∫∫Sx2yz dS, S is the part of the plane z=1+2x+3y that lies above the…

Q: 9. Find the surface area of the part of the paraboloid a = 2 – 2? – y? that lies inside the surface…

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Q: The base of a solid is the region in the xy-plane between the the lines y = x, y = 4x, x = 1 and x =…

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Q: #3. Evaluate the surface integral xz ds, where S is the part of the plane 2x +2y +z = 4 that lies in…

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Q: 4. For r > 0 and h > 0, show that the conical set S = (x, y, z) | x > 0, y > 0, z > 0, z2 h?(r2 - x²…

A: Given that the surface S is S=x,y,z|x&gt;0,y&gt;0,z&gt;0, h2=h2r2-x2-y2 We have to find the surface…

Q: 10. Find a parameterization for the part of the plane x + 2y +z = 6 in the first octant. Compute the…

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Q: Find the area of the surface generated by revolving about the y-axis the arc C given by C = { (x,y)…

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Q: Consider a sphere of radius 1 situated at the origin in R³. Find the surface area of the slice of…

A: Consider the provided question, Given that, a sphere of radius 1 situated at the origin.

Q: In exercises 13-18, compute the surface area of the surface obtained by revolving the given curve…

A: Solving part (b) as requested. Given:

Q: Find the area of the surface defined by z = xy and x2 + y2 < 7.

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Q: 1. Find the surface area of the cone z = Jx2 + y2 that lies between the plane y = x and the cylinder…

A: Surface area in xy-plane

Q: The tangent plane to the graph of the surface z=In(x2+y2) at the point (1,0,0) contains point…

A: Given function is, z=lnx2+y2

Q: Let S be the surface given by the first octant part of the paraboloid z=4-x^2-y^2. Find the area of…

A: We need to determine the surface area of the surface bounded by the paraboloid: z=4-x2-y2 in the…

Q: Let f(x, y, z) = In (x² + y + z?). The level surface f (x, y, z) = In 4 is a: Elliptical cone around…

A: Given  f (x,y,z) = ln( x2+ y+z2),To find : level surface f= ln4

Q: Given the surface of z =10-4x² – y² : Sketch the surface of z=10-4x² – y² where z>0 and show the…

A: Given:-               z = 10- 4x2 - y2z = 10-- 4x2 + y2

Q: Given a surface 3xy - In (3z - 2) = -5, where z = f(x,y). Find zx at y = 3, z = 1. Leave your answer…

A: This question is based on Partial differentiation of function

Q: Evaluate the surface integral. ∬Sz2dS,S∬Sz2dS,S is the part of the paraboloid x=y2+z2x=y2+z2 given…

A: Evaluate the surface integral. ∬Sz2dS,Sis the part of the paraboloid x=y2+z2 given by 0≤x≤4

Q: ind the area of the surface generated when the give curve is revolve about the x-axis    y=(x^3)…

A: Consider the given curve: Here the objective is to determine the area of the surface generated when…

Q: Calculate the surface area of the paraboloid with top radius R> 0 defined by 2 a² + y? 0< x² + y? <…

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Q: Consider the surface: S:z = 4 – a?, in the first octant and bounded by x² +y°, as shown in the…

A: The equation of the surface given is as follows: z=4-x2 The surface is bounded by the curve x2+y2=4.…

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

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Q: S y = 3 – 2 4z – 4 = 3(x +1)?

A: Let us start solving the problem step by step

Q: Q2 (a) Find the volume under the surface z = 3x³ + 3x?y and over the rectangle R = {(x, y): 1 < x <…

A: Volume of the surface and surface  area using double integral

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

A: Given surface is z=f(x,y)=1+2x+3y+4y2 where f(x,y)=1+2x+3y+4y2 Differentiate f(x,y) partially with…

Q: Let S be the part of the cylindrical surface x2 + y2 = 4 limited by the plans z = 0 and z = y + 2.…

A: Let S be the part of cylindrical surface x2+y2=4, limited by z=0 and z=y+2 The formula to find the…