18. Integrate G(x, y, z) = x – y – z over the portion of the plane x + y = 1 in the first octant between z = 0 and z = 1 (see the accompanying figure below). (0, 1, 1) (1, 0, 1)

Q: A fluid has density 1200 kg/m3 and flows with velocity ī measured in meters, and the components of ū…

A:

Q: Find the area of the surface x2 + y2 - z = 0 that is cut from above by the plane z = 4

A: Surface is given by x2+y2=z Area=∫∫R2x2+2y2+1dA       =∫∫R4x2+4y2+1dA Let x=rcosθ and y=rsinθ…

Q: 3. Compute the scalar line integral 3 y – xz ds where C is the line seg- ment from (0, 1, 1) to…

A:

Q: Find a point on the hyperboloid x2 + 4y2 – z2 = 1 where the tangent plane is parallel to the plane x…

A: Given  and plane

Q: D

A:

Q: Integrate G(x, y, z) = xyz over the surface of the cube cut from the first octant by the planes x =…

A: Here, the surface is in the first octant. Therefore, the integration is

Q: A fluid has density 1000 kg/m³ and flows with velocity = ci +yj + zk, where x, y, and z are measured…

A: We have to find the rate flow outward through the part of paraboloid . We can to use the methods of…

Q: d. .2 The solid bounded by the cylinder x+y = 9 and the planes z = 0 and z = 3- x. %3D

A: The solution of the problem is given in the next steps.

Q: 2. Calculate the centroid of the thin plate R which bounded by a = 1, y = 0 and, x² + y² = 4.

A:

Q: Find the sum of maximum and minimum values of the function f(x, y, z) = 6x + 12y + 18z on the elipse…

A:

Q: 2. Show that the xy plane W = (x, y,0) in Rº is generated by: (i)u = and v = (ii)u : and v = 3

A:

Q: Integrate G(x, y, z)=x4 y ( y2 + z2 ) over the surface of the cylinder y2 +z2 = 25 that lies in the…

A:

Q: 4. Use Green's Theorem to evaluate the line integral | 2xydx+ xy'dy, where C is the triangle with…

A:

Q: IntegrateF(x, y, z) = z, over the portion of the plane x + y + z = 4 that lies above the square…

A:

Q: A point moves along the intersection of the elliptic paraboloid z = a2 + 3y? and the plane y = 1. At…

A:

Q: 2. The task is to determine the flux of B across G, where G is the positively-oriented portion of…

A:

Q: Integrate G(x, y, z) = xyz over the triangular surface with vertices (1,0, 0), (0, 2, 0), and (0, 1,…

A: To Determine: integrate Gx,y,z=xyz over the triangular surface with vertices (1,0,0),(0,2,0) and…

Q: 2. Evaluate the triple in tegral where E is the y-2x2+2z20nd the plane y= 8. solid bounded by SI…

A:

Q: 3. Find the area of the part of the surface x 2z + y° - 3, that lies above the triangle in the…

A: Given, x=2z+y2-3, i.e. z=12x-y2+3. As it is known that the surface area is given as,…

Q: Calculate the line integral F(x, y, z)=2xy^2 i+(2yx^2 +2y)j     on a path from (0, 0, 0) to(3, 1, 1)

A: We need to compute the line integral: F(x,y,z)=2xy2i+(2yx2+2y)j on a path from the point: 0, 0, 0 to…

Q: The plane z =0 separates region 1 (z 0), which is a dielectric material with ɛ, = 4, from region 2…

A: The complete solutions are given below

Q: 2. Bounded by the paraboloid = 10 – 322 – 3y? and the plane z = 4 (using polar coordinates)

A: The paraboloid is bounded by z=10-3x2-3y2 . Now, converting in polar coordinates, z=10-3r2.

Q: A fluid has density 800 kg/m³ and flows with velocity i = ri + yj+ zk, where x, y, and z are…

A:

Q: 1. The solid bounded by the cylinder y = 9 – x² and the paraboloid y = 2x? + 3z².

A: In this question, we find the volume bounded by given surfaces. The surfaces: y=9-x2andy=2x2+3y2 The…

Q: 15.) The solid in the first octant bounded by the coordinate planes and the plane 3x + 6y + 4z = 12.…

A:

Q: (a) Between 1 and the disk x²+ (1+x² +y*) (b) Between % = 1 and the entire xy-plane. (1+x* + y?)*

A:

Q: 2 The values of a and b such that the surfaces ax - byz = (a + 2)x and 4x2y + z3 = 4 cut…

A:

Q: 4. (Section 17.8) Use the Divergence Theorem to calculate [[F F-dS, where F = y + ryj – zk and S is…

A: To use the divergence theorem to calculate ∫∫SF⋅dS.

Q: The tangent plane to the elliptic paraboloid z = 2x2 + y² at the point (1, 1, 3) is %3D 1) 8x + 6y +…

A:

Q: 4) Find the centroid of the plane region bounded by x² + y² = 25 and x² + (y – 6)² = 13.

A:

Q: 2. For the surface z = In(y+ Vx² + y²). (b) Find the linear approximation at the point (3,-4).

A:

Q: 4. Find the area of the surface S where S is the part of the plane z = 3x +y that lies above the…

A:

Q: Suppose that the temperature in degrees Celsius at a point (x, y, z) of a solid E bounded by the…

A:

Q: 3) Find the centroid of the plane region bounded by y = (x – 2)² and 2x – y = -4.

A:

Q: 4. Use Green's Theorem to evaluate the line integral | 2xydx+ xy*dy, where C is the triangle with…

A: The given triangle C with the vertices (0,0),(1,0) and (1,2) is as as shown below.

Q: Integrate G(x, y, z) = x - y - z over the portion of the plane x + y = 1 in the first octant between…

A: Given G(x,y,z)=x-y-z

Q: 2. Show that the xy plane W = (x, y, 0) in R³ is generated by: (i)u = 2 and v= |1| (ii)u and v = 3…

A: Given the plane: W=(x,y,0) in ℝ3.Let (x,y,0) be any element from W. For a and b in ℝ, we…

Q: (B) Integrate paraboloid (x, y, z)=- 2x-²-2-0 by the plane x=1. over the surface cut from the end of…

A:

Q: The solid bounded by z = x, z = 2 – x², z + y = 3, and y = 0

A: Volume of a solid is obtained using triple integration by the formulae =∫∫∫dx dy dz The solid is…

Q: Integrate g(x, y, z) = x4y(y2 + z2 ) over the portion of the cylinder y2 + z2 = 25 that lies in the…

A:

Q: 4. Let f(x, y, z) = e² + zy² and let C be a straight line segment connecting the points (4,2,1) and…

A:

Q: 3.4.6 Use Green's theorem in a plane to evaluate of 10x9²³- [(xy²-y) dx + (x + y²) dy] C as a double…

A:

Q: 2. Calculate the centroid of the thin plate R which bounded by r = 1, y = 0 and, 2² + y² = 4.

A:

Q: Evaluate the line integral x²z ds, where C is the line segment from (0,6, 2) to (8, 5, 1).

A:

Q: 5. Compute the flux of F(r, y.2) =zi- 2yj+ 2ak across the surface of the cylinder bounded by a+y =9…

A: Solve the problem as shown below.

Q: #1. Evaluate (2x – 5y) dA, R where R is the parallelogram with vertices (0,0), (2,3), (–1,4) and…

A: The given region R is a parallelogram with vertices as shown in below graph. We have to use change…

Q: c, y, z) = x²zi + xz'j+ y ln(x + 1) k, s the surface of the solid bounded by the 2z = 4, y = 3, x =…

A:

Q: Integrate G(x, y, z) = x2y2 + 4 over the surface cut from the parabolic cylinder y2 + 4z = 16 by the…

A:

Q: A fluid has density 1200 kg/m³ and flows with velocity v = xi + yj + zk, where x, y, and z are…

A:

Q: 10. The solid bounded by y = 4, y – x = 1, z = x², and z = = 4.

A: