2) Find the outward flux of F across the region D where F(x, y, z) = (In(x² + y²), –tan'(), zVx² + y³) and the region D is the thick-walled cylinder 1 < x² + y < 2 and –1 < z< 2.

Q: Find the flux of F¯ = (x − y)i + xj across the circle x 2 + y 2 = 4 in the xy-plane

A:

Q: (B) Integrate g(x,y,z)=x+y(x+2z) over the surface area of the region cut from plane x+2y+2z=5 by the…

A: What is Surface Integral: Surface integral is an iterated integration that is performed over a…

Q: Find the counterclockwise circulation of F = (y + ex ln y)i + (ex/y)j around the boundary of the…

A:

Q: Let S be the surface of the solid hemispherical region above the plane z = 0 and below the sphere…

A:

Q: Calculate the outward flux F*n ds where F = (x^3, y^3) and C is the boundary of the region x^2 + y^2…

A:

Q: 17. Use the Divergence Theorem to find the outward flux across the boundary of the region D where F…

A:

Q: d the volume V of the solid obtained by rotating the region bounded by the ren curves about the…

A:

Q: 5z dv, where E is bounded by the cylinder y2 + z2 = 9 and the planes x = 0, y = 3x, and z = 0 in the…

A:

Q: 1) Evaluate the line integral Fdr where F(x, y, =) = (x+3y, 2y +z, xy) and y: line segment from…

A: Here, F→(x,y,z)=x+3y,2y+z,xy and γ: line segment from (2,1,3) to (3,3,5) We need to evaluate ∫γF→dr→

Q: EXAMPLE 2 to evaluate the Use the change of variables x = u² - v², y = auv integral SSR y dA, where…

A: The objective is to evaluate the integral using change of variable. x=u2-v2, y=2uvy2=4-4x, y2=4+4x,…

Q: 1. Under the paraboloid = 3z2 + y? and above the region bounded by y = x and r= y² – y

A: Find point of intersection of:  y=xx=y2-y y=y2-yy2-2y=0y=0,2

Q: Consider a thin plate of length L and width W governed by the heat equation Uf = k(uxx + uyy) for u…

A: Let  U(x,y,t)=X(x) Y(y) T(t)ut=k(uxx+uyy)T'kT=X''X+Y''Y     ......(1) Here T is function of t. and…

Q: 39.) The solid bounded by the cylinder x+y² = 9 and the planes z= 0 and z = 3 - x. 40

A:

Q: Find the tangent plane for the level surface f(x,y,z) = 5x4 + cos (4y) + In (6z) = c at the point…

A:

Q: A flux is given by F = x²y³ i – xy j. Verify the Green's theorem for the region bounded by the…

A:

Q: Find the potential u(x, y) for u +u =0 in the rectangle 0<x< 2 and 0 < y s 2, if the upper side is…

A: The given problem is to solve the potential u(x,y) Laplace partial differential equation, we have to…

Q: Po Prove that fcz) = zis coutinuous at z=z. O Prove that. fZキ2。 is dis continnity at z=Zo, where Zn

A: (1) we have to prove that                                                              f(z)=z2 is…

Q: Let R be the region in the half plane x 2 0 bounded by the curves y=-5x + 5 y = x2 - 1 *= 0 Compute…

A: Solution

Q: 11. xyz dV, where G is the solid in the first octant that is bounded by the parabolic cylinder z = 2…

A: To evaluate the triple integral

Q: 4. Find the volume of the solid generated when the region bounded X - 2 by f(x) = x = 1, and the…

A:

Q: 4.) Verify the Divergence Theorem where F(x, y, z)=(x',xz,3z) given by x' + y +z² <4. and E is the…

A:

Q: 2. Use a line integral to calculate the mass and average density of the helix ř(t) = (2 cos t, 2 sin…

A: The given data is: r→t=2cost, 2sint, t, 0≤t≤2πδx,y,z=z To find the mass.

Q: 3. The solid bounded by z = 0, z = 1 – x², z = y and 2y + z = 12.

A: Given that  Region z=0,z=1-x^2 and 2y+z=12  To form a triple integral

Q: Use Green's Theorem to evaluate the line integral (y – x) dx + (2x – y) dy for the given path. = x²…

A: Green's Theorem: By Green's Theorem we have for a differentiable function f(x,y)=M(x,y)+N(x,y) and a…

Q: 2 2 5. Find the volume of the solid bounded by the surface f (x, y) = 3x´y´ and the curves y = 2x -…

A: We will have to find the volume of the solid bounded by the surface f(x,y)=3x2y2 and the curves…

Q: Evaluate the line integral f (-y sin(xy)i – x sin(xy)j) dr using the potential function f(x, y) =…

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: 2. Evaluate the triple integral where E is the solid bounded by y = 2x² + 2z² and the plane y = 8.

A: A detailed solution is given below.

Q: Verify the divergence theorem for F= 3i+xyj+xk taken over the region bounded by z = 4-y^2 , x=0 and…

A:

Q: 6. Use a triple integral to find the volume of a solid enclosed by the elliptic paraboloids z = 2x²…

A:

Q: Let R be the region in the half plane x 2 0 bounded by the curves y = -5x +5 y = x - 1 x = 0 Compute…

A:

Q: Consider the function f (x,y) = x3y3 + xty. Show that it is homogeneous of %3D degree 5 and verify :…

A: A function f(x, y) is said to be a homogeneous function if f(λx, λy)=λnf(x, y) where λ is a constant…

Q: Evaluate the solid bounded by 2x+z=2 and (x-1)2+y2=z.

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: The centroid of the region bounded by sin x < y< cosx, and 0<x<7/4 is, TV2-4 b) 1 1 a) 4(42 +1) 4(v2…

A:

Q: Consider the gradient flow associated to V (x, y, z) (x, y, z) = (0,0, 0) is an unstable…

A: To check whether an equilibrium point is unstable we have to find wheather that point corresponds to…

Q: You are rotating the region bounded by y 2 = x, y = 1, and x = 0 about the line x = 1. Set up the…

A: The graph formed After rotation it will look like Take the elementary element as

Q: 7. Use the divergence theorem to find the outward flux of F= 3xy²i+ xe² j+ z° k across the boundary…

A:

Q: 2. Find the coordinates of the centroid of the the region bounded by f(x) = x – x² from x = 0 to x =…

A: Question 2Here we have to find the centroid of the region bounded by f(x)=x-x2; x = 0; x = 1.First…

Q: 4. Use the Divergence Theorem to evaluate the flux integral JS, F-ndA over the surface of a…

A: Given that, F=(3xy3z5,3y,-xy3z6)F⇀=3xy3z5i^+3yj^-xy3z6k^ We have to evaluate the integral  ∫∫sF⇀.ndA…

Q: In Exercises 9–12, describe the region on which the function f is continuous. 9. f(x, y, z) = 3x²ev…

A: As per the guidelines, we are suppose to answer only one question,please repost the remaining…

Q: Find the total differential for the function w = 13z ysin x. dw = 13z ysin xdx-13z sin xdy-26z ysin…

A: Consider the function: Let,  w=13z3ysinx=fx,y,z The total differential of the function is given by…

Q: Find the tangent plane for the level surface f(x,y,z) = 6x3 + cos (4y) + In (4z) = c at the point…

A:

Q: 2. Find the volune og the solid gonned by revohing abaut thie lere y y=4x+1, y: -1 bounded by 4x +3,…

A: The volume for solid of revolution generated by revolving the area bounded by curves y=f(x) and…

Q: 8. Evaluate the line integral (x'dx+ xe"dy where C is the closed path along the triangle in the…

A: Here we have to evaluate the integral ∫cx2dx+xe-ydy where C is the closed path along the triangle in…

Q: 12. Find the outward flux of F = x*yi-cyj+zk across the boundary of the region which is bounded…

A:

Q: Verify Divergence Theorem for F = (x+ y²) î – 2 xj + 2 yzk and the volume of a tetrahedron bounded…

A: Given that F=(x+y2)i-2xj+2yzk Take x=0, y=6-2z, Then,…

Q: 24. Find the counterclockwise circulation of F = (y + e*ln y)i + (e*/y)j around the boundary of the…

A:

Q: 2. The solid in the first octant bounded by the planes x = 0, y = 0, z = 1, z = 2 – y, and the…

A: