4. Let S be the portion of the surface z = y? that lies over the rectangular region in the ry-plane with the vertices (0,0,0), (0, 1,0). (1, 1, 0) and (1,0, 0). (i) Sketch the graph of S. (ii) Evaluate the surface area.

Q: 5. Consider the surface: x(u, v) y(u, v) z(u, v) where u € [0, 1] and v € [0,2]. Calculate the area…

A: Introduction: The area is the area occupied by a two-dimensional flat surface. It has a square unit…

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

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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: Find the exact area of the surface obtained by rotating the curve about the x-axis. 2 + 2)3/2, 1s ys…

A: The curve x=13y2+232 revolved about x-axis on 1≤y≤4

Q: find an equation for the plane that is tangent tothe given surface at the given point. z = √(y - x),…

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Q: - Graph the surface 1 = y² + z² — by first sketching at least three cross sections in each of the xy…

A: To Plot: Graph The Surface 1=y2+z2-x24 by plotting at least three cross section in xy and yz plane

Q: Sketch the section of the surface at y= 0. z = V4 +x + y² 41 -8 -6 -4 -2 -4 6.

A: Let's find.

Q: B Suppose we want to find all the points on the surface gx, y) x+ 4y - 4 where the tangent plane is…

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Q: Consider the surfaces  z = x^2 +y^2   and z+ 2x+ 2y = 0 Find the projection of the curve on…

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Q: Find the points on the surface x2 - zy = 4 closest to the origin.

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Q: Consider the normal line to the surface z = x2 + y² (paraboloid) at the point P(1, 1, 2). Find the…

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Q: Find the exact area of the surface obtained by rotating the curve about the  x-axis.   x = 1/3 ( y2…

A: Given curve is x=13y2+232 over 1 ≤ y ≤ 2. The exact area of the surface obtained by rotating the…

Q: 14) Find the points on the surface xy- z2 = 1 that are closed to the origin. %3D

A: To find the point on the below surface that is closed to the origin. xy−z2=1

Q: Determine the projection of the following surface on the xy plane. (xy trace) 4z-목 + =0

A: here first solve the given surface and then we trace it

Q: 4. Graph the surface 1 = y² + z² – by first sketching at least three cross sections in each of the…

A: Graph the surface by using a 3D graphing calculator like desmos. Three cross sections in xy plane…

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: Find the shortest distance between the surface z = x² + y² and the plane r + y – z = 4.

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Q: 16)Consider the equation of the surface S given by In (xyz) + xyz + x3 - y3 + z3 = 2 An equation for…

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Q: Show that the line normal to the surface xy + z = 2 at the point (1, 1, 1) passes through the…

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Q: 1. Find the closest point of the surface ry + xz + yz = 1 to the origin. %3D

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Q: 1. Find the smallest distance from the point (0, c, 0) to the surface y = x² + z².

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Q: 12. Find the area of the surface generated when the curve y = ,on [1,2] is rotated about the x-axis.

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Q: 4. Find the area of the surface obtained by rotating the curve x²/3 + y>/3 = 1, 0 <y<1 about y-axis.

A: If the curve defined by the parametric equations x=x(t), y=y(t)  with a≤t≤b , is rotated about the…

Q: Find the unit normal to the surface z=2+2z^2-3y^3 at the point (1,-1,7), pointing down with respect…

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Q: Find the exact area of the surface obtained by rotating the curve about the x-axis. x = 1/3(y2 +…

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Q: Find the exact area of the surface obtained by rotating the curve about the x-axis. x = 1/3(y2 +…

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Q: Consider the surface given by z = 4x² + y². Draw the level curves for this surface in the (ry) plane…

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Q: 5. Find the exact area of the surface obtained by rotating the curve y = V16 x2, -2<*<2 about the…

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Q: Find the area of the surface that is generated by revolving the portion of the curve y = x2 between…

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Q: What is the equation of the surface revolved about the x-axis for z² - xz = ln x + 3x? A. y² + z² =…

A: Given:- The equation of the surface revolved for the x-axis is, z2-xz=ln(x)+3x

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: Consider the surface given by x2 + y? + z? = a². Show that every normal line to this surface passes…

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Q: 6. Find the exact area of the surface obtained by rotating the curve, y = x³, 0 < x < 2, about the…

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Q: What is the surface area of the portion of the surface z= x?+y that lays over the triangle on xy-…

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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: Find the points on surface z^2=xy+4 closest to origin.

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Q: 3. Use traces to sketch and identify the following surface 25x2 + 4y² + z2 = 100.

A: Consider the given: Given:25x2+4y2+z2=100Let x=k∴25x2+4y2+z2=100             4y2+z2=100−25x2Hence,…

Q: Let C be a curve given by the intersection of the surfaces z = x2 +y2;z = 3−2x . The value of the…

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

A: Consider a surface defined by the function z=f(x,y) over the region R. Then the area of this surface…

Q: Find the area of the surface that is generated by revolving the portion of the curve y = x2 between…

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Q: 5) Consider the surfaces S1 : ryz = 10 and S2 : z =r² + y and the %3D point Po = (1,2,5).

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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: A trace of the surface +- z = 0 on the xy- plane is: O Point at (3, 2, 0) O Circle Ellipse Point at…

A: I am going to solve the given problem by using some simple calculus to get the required result.

Q: 12. Find the area of the surface generated when the curve y =+,on [1,2] is rotated about the x-axis.

A: solution is given below...

Q: Identify and sketch the section of the surface at z = 0. v2 = 1 25 4 16 Type of the surface: too 4 4…

A: Solve the given

Q: Find the exact area of the surface obtained by rotating the curve about the x-axis.  x=1/3(y2 + 2)…

A: The given curve is,

Q: 15 - Consider the surface defined by z = f(z, y) = x³ + 2x²y +y². Find the equation of a normal line…

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Q: ii) Find the exact area of the surface obtained by rotating the curve 1 12x 3 y = x³ + 9 1 ≤ x ≤ 2,

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Q: 2. Find the area of the surface obtained by rotating the curve: y = Vx +1,1<x<5 about the Х-ахis

A: