Suppose you head toward the xy-plane from the surface x? + y? – 2² = –1 at the point (1, 1, v3) by following the normal line to the surface at that point. What are the x- and y-coordinates at which you will hit the xy-plane?

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 area of the surface generated by revolving the curve about the (a) x-axis and (b) y-axis.…

A: Explanation and solution is given below....

Q: The area of the surface obtained by rotating the curve y = 22 x + 1, 0 < x < 44 about the x-axis is

A: Consider the given curve. y=22x+1 in the interval 0≤x≤44 The problem requires to find the area of…

Q: Find the point(s) on the surface z = 3 − x2 − y2 + 6y, at which the tangent plane is horizontal.

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Q: The area of the surface obtained by rotating the curve y = 1sxse about the line x = 42 is

A: Given that y= 1/x  + x3 /12 and find the surface area. Here i attached solution of this question.

Q: Find the exact area of the surface obtained by rotating the curve about the x-axis. x3 y = 2 1 6x' 2

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Q: Find the equation of the surface of revolution generated by revolving the curve z²=4y in the…

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Q: Find the exact area of the surface obtained by rotating the curve about the x-axis. =x²+²x≤x≤1 3x 2…

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Q: The area of the surface obtained by rotating the curve y = x2, 0 <x< 12 about the y-axis is

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Q: Find the exact area of the surface obtained by rotating the curve about the x-axis. x = 2 + 4y2,…

A: To find the exact area of the surface obtained by rotating the curve about the x-axis. x=2+4y2    ,…

Q: Identify a generating curve on the XZ-plane for the surface of revolution with equation x? + y? + Z?…

A: We need to identify a generating curve on the xz-plane for the surface of the revolution with the…

Q: Determine the characteristics of the equation z = p? -q? and find the integral surface which passes…

A: Given partial differential equation is z=p2−q2 We have to find he characteristics of the equation…

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

A: Given  y =x3 ,  0 ≤ x ≤2The area under a curve is the area between a curve f(x)and the x-axis on an…

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: 2. For the surface z = In(y+ Vx² + y²). (a) Find an equation of the tangent plane at the point (3,…

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Q: Find the area of the surface obtained by rotating the curve x = /16 – y², 0 <y<2, about the y- axis.…

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Q: Find the equation of the tangent plane to the surface 2² at the point Y= 2x? (0, –1,2) %3D O y- 2z…

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Q: Suppose that an equation for the surface is S(x,y) = xye^x. In what direction (relative to the…

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

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Q: Q4) given the surface: cos nx – x²y + exz + 2yz = 2 Find the equation of the tangent plane and…

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Q: At which of the points below is the tangent plane to the surface x² – 2y + z + yz = 2 vertical?…

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Q: Find the points on the surface 2a2 + ly² + 522 = 1 at which the tangent plane is parallel to the…

A: We will use the following method for finding the two required point.

Q: 9) Find a point on the surface z = 8- 3x2 - 2y2 at which the tangent plane is perpendicular to the…

A: In this question we have to find a point on the surface at which the tangent is perpendicular to the…

Q: Find the equation of the tangent plane to x² the surface z = + y? 4 - at the point (2, 3, 10) x + 6y…

A: The equation of the tangent plane to a surface f(x,y,z) is given by:

Q: Find the equation of the tangent plane to x2 the surface z = 4 + y? at the point (2, –2, 5)

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Q: 4. Find both points on the surface xz – x2 – y? = 8 where the tangent plane is horizontal.

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Q: (b) Find the point that makes the tangent plane to the surface z = x² – xy + y? – 2x + 4y…

A: Find the partial derivatives with respect to x and y and set equal to zero. Solve for x in the…

Q: At which of the points below is the tangent plane to the surface z - 2y + z?+ yz = 2 vertical?…

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Q: Identify a generating curve on the XZ-plane for the surface of revolution with equation x? + y2 + z?…

A: Given: Surface of revolution with the equation x2+y2+z2=e2x To find: Identify a generating curve on…

Q: The area of the surface obtained by rotating the curve y = V25–x2, -4 < x s 4 about the x-axis is

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Q: Identify a generating curve on the xz-plane for the surface of revolution with equation x2 + y? + 22…

A: Solution :-

Q: The area of the surface obtained by rotating the curve y = 25-x² , -4 sx<4 about the x-axis is

A: Area of the surface rotated about x -axis is, a≤x≤b,…

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. 1 + 2x' 1 y = 1-…

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Q: 9. Show that the sum of the T-, y-, and z- intercepts of any tangent plane to the surface VI + Vỹ +…

A: Given, Gradient of the surface at any point (x0,y0,z0) is given by,

Q: At which of the points below is the tangent plane to the surface r2 – 2y? + z2 + yz = 2 vertical?…

A: From the tangent plane to a surface

Q: Find the points on the surface 2a2+ ly? + 522 = 1 at which the tangent plane is parallel to the…

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Q: Find the exact area of the surface obtained by rotating the curve x = 1/3y3, 0 &lt;= y &lt;= 1,…

A: For rotation about y-axis, the surface area of the curve is given by

Q: Find the equation of the tangent plane to the surface 9x +36y +4z? = 108 at the point (-2,1,-3).…

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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: Find the area of the surface obtained by rotating the curve y=1/33/2, 0 ≤ y≤12  about the y - axis.

A: To find the area of the surface obtained by rotating the curve

Q: Find an equation for the surface of revolution formed by revolving the curve z2 = 2y in the yz-plane…

A: To find the equation for the surface of revolution

Q: 30. The equation of the tangent plane to the surface z = x? – 2y² at x = 1, y = 0 is (а) 2 — : 2т —…

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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: x = 1+ 2y2, 1< x < 2 about the x-axis.

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Q: Find the equation of the tangent plane to the surface determined by a'y + z- 55 = 0 at a = 4, y = 3.…

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Q: Find the area of the surface obtained by rotating the curve y=1/3 x3/2, 0≤y≤12  about the y- axis.

A: Let a rectangular strip parallel to x-axis of thickness dy be made whose tip touches the curve…

Q: Find the area of the surface resulting from rotating the curve: x=5t−1, y=2t^3, -2≤t≤1, around the…

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Q: Find the equation of the tangent plane to the surface z =e/ In (2y) at the point (2, 4,1.643). Note:…

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

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