XII. $F•d$ where F = (x,y,z²) and S is sphere of radius 2 with center at 0.

Q: 15. If the surface S has projection R on the xy-plane, show that F.īdS = S da dy ה.F provided |ñ -…

A: Question: If the surface S has projection R on the xy-plane, show that ∬SF¯·n¯dS=∬RF¯·n¯dxdyn¯·k¯…

Q: The area of the region bounded by the curves 2y2 +2 and z 2y+2 is 2. 3 5 -1 -2 16 17 (A) 3 (B) 3 13…

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Q: The area bounded by the curve z 4y+7, the y-ax y =0 and y 2 is dy3D O 18 22 11 O 15

A: Area bounded by the curve can be calculated as

Q: 28. F(x, y, z) = x³i+y°j+z°k, and S is the sphere of radius 3 and centered at the origin.

A: divF→=ddxx3+ddyy3+ddzz3        =3x2+3y2+3z2        =3x2+y2+z2  by divergence theorem,…

Q: Ex.7 The area of the surface ∑ = {(x,y,z) €R® : (x² + ²) < 1, z = 9 + xy} is: (A) - 5 (B) 4개 2 (C)…

A: Using polar coordinates.

Q: 5) Evaluate SS x² yu dxdy -4 where D is the regian the bounded by the hyperbolas xy2,xyalt and…

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Q: 8. (x – y) dV, where E is enclosed by the surfaces z = x² – 1, z = 1 – x² , y = 0, and y = 2 E

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Q: 5. Find the volume of liquid needed to fill a sphere of radius R to height h (Figure 18). R h…

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Q: The area of the surface = {(x, y. z) ER :z =(x² + 3y²), x² + 9y² < 9} equals 10 O (A) T 3 O (8) 9 O…

A: Correct answer is : (D)  Explanation:

Q: The area bounded by the curve y3a and the lines z 4 and a-9 above the z-azis is aquare units O 27v0…

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Q: 17. Find the exact area of the surface z = 1 + 2x + 3y + 4y2, 1x 4, 0 y < 1

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Q: b. Evaluate the line integral ycos(xy)dx +(xcos(xy)+x)dy where R is the circle x'+y =9 traversed…

A: Solution Given b). ∫y cos(xy)dx+(xcos(xy)+x3) dyevaluate the line integral

Q: Consider the region enclosed by the curves y = a2 +1 and y = 2r + 1:

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Q: Let B be the ball with center at the origin and radius 1 SS [3(x² + y² + z²)dV =

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Q: 4 The area of the surface S = {(x, Y, z) E R³ : z = 5 – x² – y², z > 3} equals (A) 4ñ 9. (B) 5* 13…

A: 4 Given S=x,y,z∈R3:z=5−x2−y2,z≥3.. We have to find the surface area of S At z=3 ⇒3=5−x2−y2⇒x2+y2=2…

Q: (1- y*)dx + (x° + e*)dy where C is the boundary of the region between the circles x² + y² = 4 and x²…

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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: Sketch the region enclosed by the given curves. x = 9y2, x = 28 + 2y2 y y 21 35 30 1 25 20 5 10 15…

A: x=9y2,  x=28+2y2 The region is,

Q: 3. Find the area of the surface generated by revolving the curve y = 2/x ,1Sx<2, about the x-axis(as…

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Q: 2. Fnd the area of the sur face in the first octant cut from the cone z2 = x2 y2 by the plane x +y =…

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Q: 2. Find the area of the plane region bounded by y = 3x + 1, x = 1, x = 3,and the x-axis. y y = 3x +…

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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: Evaluate z² dv. where E is the region above the zy-plane below the cone z = 2+2 and inside the…

A: The given problem is to evaluate the given triple integral over the given region E bounded above by…

Q: c) Evaluate the line integral (x+2y)dx -2xdy along the following closed path, the circle x+ y= 25,…

A: In parametric form,

Q: A surface x(y, z) is defined implicitly xy + yz +x'z' = 3-xZ Calculate ду

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Q: # 4. Evaluate the line integral | (x +2x²y) ds, where C is the upper half of the unit circle x2 + y2…

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Q: 4. Evaluate the line integral / r y* ds where C is the right half of the circle r² + y² = 16.

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Q: hy sphere x + y +z = 25 between planes z=2 and z= 4.

A: We need to find surface area using double intergral.

Q: 7. If F = (yz,-2xz,xy), the value of ||F.n dS , where S is a sphere of radius a centered at the…

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Q: Evaluate || F.ndS F = xi + yj+(z² –1)k if and o is the curved surface of the cylinder y = \4-x? from…

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Q: Find the area of the band cut from the paraboloid x² + y² – z = 0 by the planes z = 2 and z = 6.…

A: Given equation is, x2+y2 = z Band cut from the given paraboloid by the planes z = 2; z = 6 Area of…

Q: 6- Evaluate the line integral fF dr, where F = [xy, yz, xz] and C is the boundary of the hemisphere…

A: the line integral f F dr, where F = [ry, yz, xz] and C is the boundary of the hemisphere x² + y² +…

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: The area enclosed between the curves p(a) =2 +*+ 3 and g(@) = -22 - * + is O 16 47 %3D 18 49 O -49…

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Q: Find the extreme values of the function f(x, y, z) = xy + z^2 on the circle in which the plane y - x…

A: Given function is fx,y,z=xy+z2. We have to find the extreme values of the given function on the…

Q: 135 2. y = x³, y = x + 6, and 2y + 2 = 0 (Answer: s. u.) 4

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Q: 4 The area of the surface S = {(x, y, z) € R³ : z — 5 — z² - y², z ≥ 3} equals (A) 4T (B) (C) (D)…

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Q: Evaluate ∬S(x2+y2)dS, where S is the surface z=4−x, 0≤x≤2, 0≤y≤2.

A: Given z=4-x , 0≤x≤2 ,0≤y≤2 to evaluate ∬Sx2+y2dS where S is surface

Q: 4. Evaluate ||| x y dV , where E is bounded by the parabolic cylinders y = x², y' = x and the planes…

A: The region E is bounded by the parabolic cylinders y = x2, x = y2 and the planes z = 0 and z = x +…

Q: (b) Let F = (-yx², xy²). Let D be the annulus 1 < a2 + y² < 4, and let C be its boundary. Evaluate

A: Value of the given integral is

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

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Q: 3. Let D be the region bounded by the parabola y= -3 and the line y=1.F

A: The region is bounded by the curves given by: y=x2-3      and      y=1

Q: Find Ja xydx + sin(x)ydy, where C is the boundary of the triangle with vertices (0,0), (0, 1) and…

A: Let I is the value of given Line Integral.

Q: d). Integrate f(x, y, z) = -√√x² + z² over the circle r(t) = (a cost)j + (asint)k, 0≤t≤2.…

A: Solution:-

Q: B. Let C be the curve y = 1 – x2, 0< x < 1. If C is rotated about the y axis find the area of the…

A: our objective is to calculate the surface area.

Q: 33. Figure 14.13 shows the saddle-shaped surface z = S(x, y). (a) What is the sign of f,(0, 5)? (b)…

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Q: On the surface z =, in what direction is z increasing most rapidly when x =1 and y = 2.

A: A function z=f(x,y) increases most rapidly, at a point (a,b) , in the direction of the gradient…

Q: 1. Find So(r+2y)dA, where D is the region bounded by the parabolas y r, and y=1+

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Q: Sketch the reglons enclosed by the glven curves. y = Vx, y= X= 16 y y 6 4 4 3 3 2 10 15 10 15 O-1…

A: Given : y = √x  y = 1/3 x  x = 16