1. a) Evaluate I = S S Sp(x² +y² – 1) dV, where E lies between the spheres a2 + y? + 22 = 9 and a? + y² + z² = 16. b) Which coordinate system did you choose to evaluate integral I? Explain why.

Q: 1.10.6 Show, by expansion of the surface integral, that S, do x V lim = V x V. sdt-0 fdr Hint.…

A: To show that lim∫dτ →0∫sdσ×V∫dτ=∇×V

Q: 1 Evaluate E 22 + y² + z² dV, where E lies between the spheres a? + y? + z² = 16 ar ² + y² + z² = 25…

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Q: 1. Use Green's theorem in order to compute the line integral $ (3cos + 6y?) dx + (sin(5") + 16x³) dy…

A: Here the given integral is                               ∮C3cosx+6y2dx + sin5y+16x3dy

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Q: 46.) (a) Parameterize S the 1st octant portion of the cylinder y² + z² = 9, 0 <x<4 (b) Use above to…

A: y2+z2  0≤x≤4x=uy=3 cos vz=3 sin vparameterise the surface ass(u,v)=(u   3 cos v   3 sin v)0≤u≤4,…

Q: Use spherical coordinates to evaluate the triple integral x² + y² + z² dV, where E is the ball: x² +…

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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: Consider the triple integral I = S S“(y² – 6z)dydzdx. -

A: ∫02∫x21∫0xzy2-6zdydzdx=∫02∫x21z3x33-6z2xdzdx=∫02x31-x812-6x13-x63dx=31.88888…

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A: We have to find the value of given line integral.

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

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Q: 1. Evaluate SSSE z dV where E lies between the spheres x2 + y? + 2² = 1 and a? + y² + z² = 4 in the…

A: Given, ∫∫∫EzdV where E lies between the spheres x2+y2+z2=1 and x2+y2+z2=4 in the first octant.

Q: 2. a. Calculate the surface integral of /(x.y,z) = obtained by parametric equation: over the full…

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Q: Evaluate ſ[ſp z dV where E lies between the spheres x² + y² + z² = 1 and + y*¨+ 2² = 4 in the first…

A: Solution:

Q: 1 Evaluate dV, where E lies between the spheres x + y + z = 16 and + y? + z2 = 64 in the first…

A:

Q: 1 Evaluate dV, where E lies between the spheres a? + y? + z? 25 and + y² + z? x² + y? + z? = 36 in…

A:

Q: Evaluate tne following double integral ! VI- 2 (2 + y)* dyd.r. -1 0

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Q: ¹ // x² + y² + 2² dV, where E is the ball: x² + y² + z² ≤ 16. Use spherical coordinates to evaluate…

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Q: (22. //I (a² + y²) dV where E is the solid that lies between 2 spheres x² + y? + z2 = 4 and a? + y²…

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Q: Y – 5x |dA x + y 100 cos

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Q: 10.18. Use an appropriate change of variables to evaluate the integral ff,(1 + x? + y²)²dA, where T…

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Q: 6. Set up the integral /| f(x, y, 2) dV E where E is the solid that lies under x + 4y + 6z = 12 in…

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Q: 1. Evaluate the integral: SSf, xe**+y<+zt dV where B is the portion of the unit ball x² + y² + z² <…

A: Here, the region B is the portion of the unit ball x2+y2+z2≤1 that lies in the first octant.…

Q: Evaluate the triple integral. 6xyz dV, where T is the solid tetrahedron with vertices (0, 0,0), (1,…

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Q: 10. Let D be the solid inside the cylinder r= cos , above the xy-plane and under the paraboloid…

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Q: 3) Evaluate the cylindrical coordinate integrals in: 2n 1 1/2 ISPsin'e + z") dz r dr de e-1/2

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Q: (1) Sketch the regions of integration. (a) 3m/2 (b) (r,0)r drdo 3m/4

A: The integrations  (a). ∫-π2π2∫04f(r,θ)rdrdθ(b). ∫3π43π2∫34f(r,θ)rdrdθ We have to sketch the regions…

Q: Evaluate the triple integral [ 2³ dV, where E is the solid tetrahedron with vertices (0,0,0),…

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Q: Calculate the value of the double integral 2 c3 5(x'y+ xy' )dædy. Ja=0 Jy-0 O 70 O 210 O 190 O 150

A: Consider the given double integral. ∫x=02∫y=035x2y+xy2dxdy First integrate with respect to y.…

Q: ¹ [ ] [ ²0 x dxdydz where E is the solid limited by Plane E z = 0,z=x+y+5 and by the cylinders x² +…

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Q: Solve the DE, dy =y²-a? E, = , using the homogeneous method. ry (a) Hint: Rewrite the RHS as 2 -…

A: dydx=y2-x2xy⇒dydx=y2xy-x2xy⇒dydx=yx-xy⇒dydx=yx-yx-1 Now put, y=vx⇒dydx=v+x·dvdx

Q: Let 0 < c < 1. Compute the integral rc+ioo dz Jc-ioo sin nz' where the integration path is the…

A: We will use the basic knowledge of complex line integrals to answer this question properly, in…

Q: The integral JJI eV 4x2 + 4y² + 4z² dV = mt, where R is the solid hemisphere with x2 + y? + z² < 11…

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Q: - Evaluate the integral 4-z²-y² 1 dz dy dx Jo 4-z²-y² x² + y² + 2² using spherical coordinates.

A: To evaluate ∫-22∫04-x2∫-4-x2-y24-x2-y2 1x2+y2+z2 dzdydx

Q: Consider the solid lih S,: X=4-y² そ=2

A: Surface: Surface, A two-dimensional collection of points (flat surface), a three-dimensional…

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

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Q: 1. Evaluate ſf, y dS where S is the portion of the cylinder 4x² + 2y² = 16 that lies between z = 0…

A: Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Evaluate the triple integral II. x* dV, where T' is the solid tetrahedron h vertices (0, 0, 0),…

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

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Q: Use either form of Green's Theorem to evaluate the following line integral. 03x y dx+ 4x y dy; C is…

A: According to the given information, it is required to find the value of integral using Green’s…

Q: 3) Evaluate the cylindrical coordinate integrals in: 2n 1 1/2 || | (r²sin²0 +z²) dz r dr d0 0 0 -1/2

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Q: And And where What is the produc A C sdR = dxi + dyj + dzk 3 Ã = zi – xj – yk Cya S. Ā.dR of…

A: Ans = 11/2  For detail go through the solution

Q: need to find the limits of integration in terms of ρ, φ, and θ.

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Q: 2 Sphere S to be the top half of sphere · x² + y² + 2² SS₁zdS /5 calculate the integral.

A: The given Surface is the top half of the surface x2+y2+z2=a2. To Evaluate: ∬zdS.

Q: 1. Evaluate the integral: SSS, x² + y² + z² dV where D lies above the cone z = spheres x? + y? + z²…

A: Here we have to evaluate the integral value, where D lies above the cone z= √(x2+y2) and between the…

Q: 1. Evaluate ſſ, y dS where S is the portion of the cylinder 4x² + 2y² = 16 %3D that lies between z =…

A: Given S is portion of cylinder 4x2+2y2=16 that lies between z=0 and z=7. Parametrize the surface s.…

Q: 3. Solve the Laplace equation inside a 60° wedge of radius 1 subjected to the boundary conditions…

A: The Laplace equation is given by: r2∂2u∂r2+r∂u∂r+∂2u∂θ2=0 The boundary conditions are: ∂u∂θr,0=0,…

Q: Set up but do not evaluate the triple integral fff ex+v+dV, where S is the solid lying below S…

A: Given S is a solid lying below z+2x2+2y2=9, z=1 and outside the cylinder x2+y2=1  I=∫∫∫ex+y+z dV

Q: 17 z cos(8 z) dz Which of the following choices of u and du are consistent with the principle of…

A: Solve the integral using integral byparts

Q: the iterated integral Evalua te 2-y 14-y" (~zdxdz dy 2 0.

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:…