Example 6: Evaluate ez dz dy dx E Where E is enclosed by the paraboloid Z = 3+ x? + y² , the cylinder x? + y? = 2 and xy plane.

Q: 1. The solid that is above by the paraboloids z = 2- x² – y² and below by the paraboloids z = x? +…

A:

Q: In AMNP, T is the centroid. If MR=30 find MT. N R T M S

A: We are given that T is the centroid and MR=30. We know that the centroid divides each median in a…

Q: 8.Evaluate the triple Integral. Te by The paraboloid X=2yt + 2z? and the plane X=2. 7x dV, where E…

A:

Q: Evaluate the triple integral |// xy dV where E is the solid tetrahedon with vertices (0,0,0), (7, 0,…

A: Consider the following: Equation of plane passing through 7,0,0,0,5,0,0,0,10x7+y5+z10=1x ​ranges…

Q: Question 3. Compute the integral dz 8+z where y is the rectangle with vertices +3+i with the…

A:

Q: Find the Uolume of the Solid under de the 2:1+ xy2 & the a Gove te region Sorface en cased by y= x²…

A: Consider the given surface z=1+x2 y2 Since, y=1 and y=x2 Then, x=±1 That is, -1≤x≤1  and    x2≤y≤1

Q: Evaluate e dV where E is enclosed by the paraboloid z = 2 + x² + y°, the cylinder z² + y² = 2, and…

A:

Q: 1. a) Use the cylindrical coordinates to evaluate (3 + xy)dV, where E is the solid E in the first…

A: Here is the solution of the given problem.

Q: Evaluate f (x + 2y)dx + (y – 2x)dy around the ellipse C defined by x = 4 cos 0, y = %3D | C 3 sin…

A:

Q: Fined Fhe boundled by Ahe Cure area 2. x+Y= 3 %3D -33-j` and ど= %3D Ans wer

A:

Q: pプ PG2-2(6x)さ子エ theorem over the elosed hemispher defined bY Zz J25-x²_y?

A: Given that F→=x2+yi^-zyj^+z2k^ and z=25-x2-y2

Q: The parametric form of the solutions of the PDES Uy + u u z = 0 , u(x,0) = -x is

A:

Q: Evaluate the triple integral∭E xy dV where E is the solid tetrahedon with vertices…

A: Given Data : Triple Integral ∬∫ExydV and E is the solid tetrahedron with vertices 0,0,0,…

Q: Find a solution to the following Dirichlet problem for a half disk. Zu 1 du 1 Pu = 0, 0<r<1, 0<0<n…

A:

Q: Find a solution to the Dirichlet boundary value problem for a disk. 2u 1 du 1 Pu = 0, r dr r2 a02 +…

A: Answer is mentioned below

Q: Considering f(x, y, z) = z + x²y and S as part of the cylinder y? + z² = 25 that is between the…

A: Given that, fx,y,z=z+x2y and S is the part of the cylinder y2+z2=25 that is between the planes x=0…

Q: . Solve the following Dirichlet problem on a disk V²U(r,0) = 0, 0<r<1, -asosa U(1,0) = 0, - <o< T.

A: This question based on Dirichlet problem on a disk from partial differential equation.

Q: Problem 52. Use Stoke's theorem to evaluate fy°dx – x³dy + z³dz where C is the trace of the cylinder…

A: Given integral and curve are,

Q: Evaluate // e² dV where E is enclosed by the paraboloid z = 2 + x² + y², the cylinder E x² + y? = 3,…

A: Since we are given with the multiple number of questions and answered according to the guideline.

Q: Evaluate the triple integral xy dV where E is the solid tetrahedon with vertices (0, 0,0), (6, 0,…

A:

Q: • SSS ²³² x² + y² = 2, and the xy plane. Evaluate e² dV where E is enclosed by the paraboloid z = 6…

A:

Q: Find a solution to the Dirichlet boundary value problem for a disk. zu 1 du 1 Pu = 0, + -- ar? Osr<…

A:

Q: [y sxy dy is equal to -y cony - J cony A (B у сoly - coty dy Scory dy C у coly + -y coty + J coty D…

A: Solution is given below..

Q: 39. Show that the graph of r =t sin ti + t cos tj + t'k lies on the paraboloid z = x² + y².

A: Here we compare r as its basic equation to prove this.

Q: By using Green's Theorem, evaluate the integral (1+xy') dx +x²y dy, where R is the closed R path…

A: From the given problem:   I=∫R1+xy2dx+x2ydy   From Green's theorem:   ∫F1dx+F2dy=∬∂F2∂x-∂F1∂ydxdy…

Q: Use either form of Green's Theorem to evaluate the following line integral. $9x²1² 6 C $9x³y² dx +…

A:

Q: Find z dV, where E is the solid tetrahedron with vertices (0,0,0), (1,0,0), (0,2,0), and (0,0,4)

A: x varies from 0 to 1&amp; y varies from 0 to 2 and z varies from 0 to 4.

Q: 1. Evaluate fL, 2 dv where E lies below the plane z = 2+ 3x + 4y and above the region in the…

A:

Q: 5. Evaluate y* dV, where E is the solid tetrahedron with vertices at (0,0,0), (2,0,0), (0, 2,0), and…

A: we can evaluate the given integral. ∫∫∫R f(x,y,z) dV =∫∫∫R f(x,y,z) dzdydx

Q: Evaluate the triple Integral. 3xyz dV, where T

A:

Q: Find z dV, where E is the solid tetrahedron with vertices (0,0,0), (1,0,0), (0,3,0), and (0,0,5)

A:

Q: 3. Evaluate /// 2* dV if G is a solid in the first octant bounded by the plane y + z = 2 and the…

A: triple integral

Q: Solve the BVP for the disk (exterior Dirchlet problem) v²u = 0, 1< r <∞ u (1,0) = sin 0 + cos 20

A:

Q: Question 3. Compute the integral - dz 8+z where y is the rectangle with vertices +3+ i with the…

A:

Q: 1 Evaluate dV, where E lies between the spheres x2 + y2 + z2 = 9 and x2 + y? + z2 x2 + y? + z2 36 in…

A: we want to find the triple integral ∫∫∫E1x2+y2+z2  dVwhere E lies between the sphere x2+y2+z2=9  and…

Q: 1 I. Evaluate dV, where E lies between the spheres x? + y² + z² = 1 and x2 + y? + z² E x2 + y? + z?…

A:

Q: Evaluate p(10ey – 9x)dy – (yx – 5 In x)dx by using Green's Theorem where C is a trapezium with…

A:

Q: b) a dr+(x+yz) dy+(ry- VE) dz, C is the boundary of the part of the plane 3r+2y+z in the first…

A:

Q: Evaluate the triple integral.  (x2 + y2) dx dy dz; W is the pyramid with top vertex at (0, 0, 2)…

A: The given triple integral is,

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: Evaluate the triple integral III xy dV where E is the solid tetrahedon with vertices (0, 0, 0),…

A: The given integral is, ∫∫∫E xydV Here E is the solid tetrahedron with vertices O0,0,0, A10,0,0,…

Q: 1. Draw a sketch of the cylinder z= ==V36 – 4y2.

A: Given: z=1336-4y2

Q: Evaluate y* dV, where E is the solid tetrahedron with vertices at (0,0,0), (2,0, 0), E (0, 2,0), and…

A:

Q: Evaluate e dV where E is enclosed by the paraboloid z = 4 + x² + y², the %3D cylinder x? + y² = 3,…

A: We need to evaluate integral.

Q: 1. Evaluate the integral ryz dV, where E is the tetrahedron with vertices (0.0,0), (2,0,0), (2,3,…

A: The vertices of tetrahedron are (0, 0, 0), (2, 0, 0), (2, 3, 0) , (2, 0, 4). The equation of the…

Q: 1 9. Evaluate dV;where E is the solid between the spheres of radius 1 (x² + y² + z²)³/2 E and 2…

A:

Q: 1. Evaluate the line integral I = = [v¢4x° + y*) dx + x(2x° + 3y*) dy] around the ellipse x/a? +…

A: The given line integral is, I=∮y4x2+y2dx+x2x2+3y2dy We know the Green's theorem:…

Q: Solve the BVP for the disk (exterior Dirchlet problem) v'u = 0, 1< r <o∞ u (1,0) = sin 0 + cos 20

A: The given function is ∇2u=0 The given condition is u(1,θ)=sinθ+cos 2θ We have to find the general…

Q: Example 5: Find fff, Vx2 + z²dV, where E is the region bounded by the paraboloid y = x2 + z2 and the…

A: Cartesian co-ordinate and polar co-ordinate x=rcosθy=r sinθx2+y2=r2θ=tan-1yx

Q: 1. The accompanying figure shows two polygonal paths in space joining the origin to the point (1, 1,…

A: Given: The function with two polygonal paths in space joining the origin to the point (1, 1, 1).…