Consider the oriented surface S for z > 0, consisting of the portion of the surface of the paraboloid z = 4 – (x² + y²) above the xy-plane and with outward normal. (a) Please sketch the surface S. (b) What is the boundary curve C = aS and what direction is its positive orientation?

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A: we have to find the equation of the tangent plane of the surface 2=(x+1)2+(y-2)2-1 atthe point…

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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: Consider the surface z = 2 – xy in R³. Use the second derivative test to find the points on this…

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

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Q: Consider the surface z = x In y – x²y. (a) Show that Q = (1,1, –1) lies on this surface. (b) Find…

A: Consider the surface: z=xlny-x2y…

Q: Consider the oriented surface S for z > 0, consisting of the portion of the surface of the…

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Q: If (a,B,y) is a point at which the surface x-+y²-z²-2x+12=0_has a horizontal tangent plane, then…

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Q: Find the area of the following surface using a parametric description of the surface.   The cap of…

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Q: Find the exact area of the surface obtained by rotating the curve x = (1/3)y^3, 0 &lt;= y &lt;= 1,…

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Q: Consider the surface 4a? – y? + 2z² + 4 = 0. a. Find the value(s) of k such that the cross-section…

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Q: Sketch the section of the surface at y = 0. 1 + x² + y² Z = 4 -8 -6 -4 -2 4 4 6. 4 -8 -6 -4 -2 2. 4…

A: Given data is: To sketch the surface at y=0 z=1+x2+y2

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Q: Find the equation of the surface of revolution of x = z2 about the x-axis.

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Q: Find the exact area of the surface obtained by rotating the curve about the x-axis 극v2 + 2)3/2, 4…

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Q: Find the exact area of the surface obtained by rotating the curve about the x-axis. 1 + 2x' 1 y = 1-…

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Q: Let S be the surface with equation (z - 4)² = 4x² + 4y². Viewing S as a surface of revolution, find…

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Q: Let S be the surface obtained by revolving the plane curve 2x = /9 – y² about the y-axis.

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Q: Let f(x,y)=/4– 4x² – y² . Find its domain and give a rough sketch of its surface.

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Q: Let C be the curve y =x3/2, 1 < x < 4. (a) Find the length of this curve. (b) Find the surface area…

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Q: Let C be the curve y = 6x³ for 0 ≤ x ≤ 1.3. 13- 12- 11 10- 9- 8- 7- 6- 5- 4 3- 2- 0.5 a Find the…

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Q: For the surface a?+ - z = 1, a curve in a form of a parabola can be formed bi passing through a…

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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 x² - y² + z² = −4. A. Identify the name of the surface. B. The line…

A: We know general form of hyperboloid of two sheets is  -x2a2-y2b2+z2c2=1 where positive term is the…

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Q: (b) Find the point that makes the tangent plane to the surface z = x- xy +y-2x + 4y horizontal.

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Q: Given the surface z = In(-2x² + 5y² – xy). Find the equation of the tangent plane and the parametric…

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Q: 3. Find the surface area of the surface generated by revolving the graph of the "curve" given…

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

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Q: Consider the surface given by the equation z - y = x2. Draw a total of at least 3 traces in at least…

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Q: Let S is the first-octant portion of the plane 2X+Y+Z=4. Evaluate the surface integral ∬(2Y^2+2YZ)…

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Q: Give an equation of the surface of revolution generated by revolving x In z about the z-axis.

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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 + 2)…

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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: Consider the surface z = 4x2² + y² + 1. An equation for the tangent plane to this surface at the…

A: Given: z=4x2+y2+1

Q: The area of the surface obtained by rotating the curve x = y2 - In(Vy), 1sys2 about the x-axis is

A: The given curve is: x=14y2-lny  ;      1≤y≤2 Therefore: dxdy=y2-12y

Q: The exact area of the surface obtained by rotating the following curve 6x = y + 12, 2 < x < 3 about…

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