Compute sF · dS for F = (y², 5, x), S being the portion of the plane x + y + z = 1 in the octant x, y, z > 0, with an upward-pointing normal. (Use symbolic notation and fractions where needed.) F. dS =

Q: Q1: Evaluate: 2x dV where E is the region under the plane 2x + 3y + z = 6 that lies in the first…

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Q: Let F(x, y, z) = (xz, yz, z²). (a) Let S be the cylinder a2 + y² = 4, 1 < z < 3, oriented by the…

A: Hello. Since your question has multiple questions, we will solve first question for you. If you want…

Q: Given the curve y = x´ and the lines y = 0 and y=2 – x Find the moment of R with respect to the…

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Q: Find the normal line for the level surface f(x,y,z) = 3x3 + cos (2y) + ln (2z) = c at the point…

A: f(x,y,z)=3x3+cos(2y)+ln(2z)-c=0Fx=9x2 , Fy=-sin(2y) , Fz=12z×2=1z∇F =&lt;Fx,Fy,Fz &gt;=&lt;9x2,…

Q: a) Evaluate // (-xi- yj+2z k) · n dS, where o is the portion of the paraboloid z = 3x2 + 3y² between…

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Q: Calculate F dS if F(x, y, z) = x²yei+ xyj – 2xye k, where S is the surface of the pox bounded by the…

A: Given-F(x,y,z)=x2yezi+xy2j-2xyezk To find- ∬SF.dS ,where S is the surface of the box bounded by the…

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Q: integrate ƒ over the given region. ƒ(u, y) = y - sqrt(u) over the triangular region cut from the…

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Q: Find the normal line for the level surface f(x,y,z) = 4x² + cos (3y) + In (5z) = c at the point…

A: The given surface is: f(x,y,z)=4x2+cos(3y)+ln(5z)

Q: Find the polar moment of inertia about the origin of a thin triangular plate of constant density d =…

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Q: Find the average value of f(, y) = 2ry over the rectangle with vertices (0, 0), (5, 0), (5, 3), (0,…

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Q: Find the normal line for the level surface f(x,y,z) = 5x3 + cos (5y) + In (6z) = c at the point…

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Q: Find the normal line for the level surface f(x,y,z) = 3x5 + cos (5y) + %3D In (5z) = c at the point…

A: Question is solved.

Q: Find the normal line for the level surface f(x,y,z) = 7x³ + cos (4y) + In (4z) = c at the point…

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Q: Find all points on the surface z = 2 + xy at which the normal line is parallel to the line x = 3 +…

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Q: 4. Locate the centroid (x, y)of the following composite areas: -3 ft 3 ft I 1.5 ft 1 ft

A: Consider the given composite area and divide then into four areas as shown in diagram below,

Q: Find all points on the surface x² + y² – z² = I at which the normal line is parallel to the line…

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Q: b) Use Gauss theorem to evaluate F.ndS, where F(r, y, 2) = (2x²y² + In z) i – xy³ j – tanh(a² + y³)…

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Q: 2. Let S be the surface x² + y² = (2-2)², y Evaluate Jl F = (2x+y-xz, -x, -(2 — z)²), choosing a…

A: Using parametrization of the surface S

Q: Compute ,F . dS for F = (y, 5,-x), S being the portion of the plane x + y + z = 1 in the octant x,…

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Q: Let F = 1xi + 5yj and let n be the outward unit normal vector to the positively oriented circle x² +…

A: To find the flux we have to first compute unit normal vectors by using formulan=Gradient…

Q: The area of the region defined by Il æ| – |y|| < 1 and a + y < 1 in the xy plane is a. T b. 27 С. Зт…

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Q: Find the normal line for the level surface f(x,y,z) = 6x³ + cos (4y) + In (4z) = c %3D o a. r(t) =…

A: Given that: f(x, y, z)=6x3+cos(4y)+ln(4z)=c

Q: Let n be the outer unit normal of the elliptical shell S: 9x +25y +225z = 225, z20, and let %3D F=…

A: ∇×F=ijk∂∂x∂∂y∂∂zyx2x2+y432sin…

Q: Let F(x,y,z)= (6ye?,2e?,12(2-e?)). Let S be the portion of the graph z=In(6x2+120y3+2) which is…

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Q: Let F(x, y,z) = (-xe² +3x, -ye² + arctan(x? + z), 2e²) and S the portion of the sphere acoording to…

A: Given that Fx,y,z=-xez+3x, -yez+arctanx2+z2, 2ez. Also, S=x,y,z| x2+y2+z2=1, z≥0. To determine the…

Q: 10. Find the area of the paraboloid x² + y + z² = 2 by the plane y = 0. (Hint: Implicit Surface)

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Q: 4) Use Stokes theorem to evaluate SF.dr, where с F(x, y, z) = <.xy, yz, *x), where C is the triangle…

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Q: Let (S) be the surface given by the following equation. In(2x + y) = ze* + 3y². An equation of the…

A: Solve the following

Q: Find a unit normal vector to the surface f(x, y, z) = 0 at the point P(4, – 5, – 5) for the fur 5x…

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Q: 2. Evaluate [(3yz)dV where B is the region in the first octant lying below the plane x + 2y +2: = 4.

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Q: Calculate the points of intersection of the curves x² = 4y and bn=) = y 29vaus at newed biloa gris 1…

A: Centroid: The centroid or centre of mass of a region bounded by the curves is the point where the…

Q: Let T be the triangular region bounded by the lines x + y = 1, x = 0, and y = 0. Evaluate the line…

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

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Q: Compute the surface integral of the normal component of F(x,y, z) = xi over the triangle with…

A: According to the given information, it is required to calculate the surface integral.

Q: Find the centroid of the third quadrant area bounded by the following curves y2+ 2y + 8x = −1 and y…

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Q: Find the points on the surface xy + yz + zx - x- z =0 where the ant pla

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Q: Compute ,F · dS for F = (xy, 5y,0), S being the cone z² = x + y²,x² + y? 0, with the normal %3D…

A: The vector field is F=⟨xy, 5y, 0⟩ and the parametric equation of the cone is  Rr, θ=rcosθ, rsinθ, r…

Q: E is the region behind the surface y = 4 – x² that is in front of the region in the xy-plane bounded…

A: Given integral: ∫∬E18x dV Given that E is the region behind the surface y=4-x2 that is in front of…

Q: Let F(x,y,z)= (4ye?,2e?,12(2-e?)). Let S be the portion of the graph z= In(4x2+60y3+2) which is…

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Q: 303. Compute F. NdS, where F(x, y, z) = xyi + zj + (x + y)k and N is an outward normal vector S,…

A: According to the given information, it is required to compute the integral where S is the triangular…

Q: 2. Suppose that S is a surface in R3 with unit normal vector N surface area of S is given by (n1,…

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Q: Find an expression for x, the x- coordinate of the centroid of the region bound by the curve x = 9 -…

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Q: Find the average distance from a point P(x, y) in the disk x2 + y2&lt;=a2 to the origin.

A: Average distance is given as

Q: Verify Green's theorem in the plane for (3x² -8y²)dx+(4y-6xy)dy, where C is the closed curve of the…

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Q: Calculate / F . dS where F = (e* sin y, e cos Y, yz²) S is the surface of the box bounded by the…

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Q: Find SS F dS where F(x, y, z) = z i+(3x – 1)j+ yk and S is the portion of the plane x + y + z = 1…

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Q: Find the normal line for the level surface f(x,y,z) = 7x5 + cos (5y) + In (9z) = c at the point…

A: The given level surface is: f(x,y,z)=7x5+cos(5y)+ln(9z)Given point is:  (9, 3, 10)

Domain

In simple mathematics terms, a domain is defined as the space on which all the possible inputs for the function will lie. It can also be expressed as a set of departures for the required function.