3. Express the integral I = f(x, yz) dV as an iterated integral in three different ways, where E E is the part of the spherical solid x² + y² + z² ≤ a² in the first octant. (a) In rectangular coordinates dz dy dx. (b) In cylindrical coordinates dz dr de. (c) In spherical coordinates dr do do.

Q: 2 ² J z=0x= -√4-2² 4-x²-2 y=0 N dydxdz

A: The given integral is: ∫z=02∫x=-4-z24-z2∫y=04-x2-z2 z dy dx dz Convert the given integral into…

Q: Evaluate the cylindrical coordinate integrals

A: To evaluate: ∫02π∫0θ/2π∫03+24r2  dz r dr dθ

Q: Compute the differential of surface area for the surface S described by the given parametrization.…

A: Given that we need to determine the differential of surface area using above parametriation.

Q: 2. Let F = −y³i + x³j + cos z³k and C be the curve generated by the intersection of the cylinder x²…

A: a. Given, F¯=-y3i¯+x3j¯+cos z3 k¯ Let C be the curve generated by the intersection of the cylinder…

Q: 2. Given F(x, y, z) = -y²ī+ xj + z²k and the curve C, intersection of the plane y + z = 3 and the…

A: Given that F→(x, y, z)=-y2i+xj+z2k and the curve C, intersection of the plane y+z=3  and the…

Q: (5) Use Stokes' Theorem to find the line integral |(* + 2y°)i + (y + z°) +(= + 2r*)k) • dř where C…

A:

Q: Write a triple iterated integral in cylindrical coordinates (dzdrdtheta) and solve

A:

Q: Use cylindrical coordinates to evaluate d V when

A: The given integral is ∭RxdV where the region R is defined by R:x2+y2≤1, x≥0, y≥0, 0≤z≤2. Converting…

Q: luate the surface integral. // (x + y + z) ds, S is the parallelogram with parametric equations x =…

A: Sub x= u+v  y= u-v  z= 1+2u+v

Q: Evaluate the iterated triple integral using cylindrical coordinates 4-r²-y² √3 √4-1² 4-x²-y² fer: dz…

A: As we have to use cylindrical  coordinate system . So we need to convert cartesian into cylindrical…

Q: (xy + z) dz dx dy into an integral in cylindrical coordinates. Convert the integral (D); ,(0). (O)…

A:

Q: Compute the volume of the solid below 2 – a? – y? and above the region D in the xy-plane. The…

A:

Q: Use Stokes' Theorem to find the line integral [((x² + 2y² ((x + 2y²)i + (y + 2²)j + (2 + 2x²)k) · dř…

A: Given that C is boundary of triangle with vertices 1,0,0,0,1,0,0,0,1 So, equation of the plane with…

Q: 3. (15 pants) Evaluate the line integral Y ds where C is part of the parabola y = x² from (0,0) to…

A:

Q: Use spherical coordinates to find [[[, z2 dV.

A: In this question, we find the triple integral by using spherical coordinate with a sketch of solids

Q: 2. The vector function Ř(u, v) = (sin u cos v, sin u sin v, cos u), (u, v) E [0, 7] × [0, 2m] gives…

A: Here, the parametric vector function for the surface is given by R→u,v=sin u cos v, sin u sin v, cos…

Q: Compute the volume of the solid below 2 – a? – y² and above the region D in the xy-plane. The…

A:

Q: Verify the line integral and surface integral by relating them to the Stokes' Theorem where C is the…

A: Stokes theorem    ∫CF→.dr→=∫∫ScurlF→.dS→ Where S  be a smooth surface  bounded by a closed, smooth…

Q: Convert the integral 16-x2 3-y I L to an equivalent integral in cylindrical coordinates. Do not…

A:

Q: Evaluate the integral by changing to cylindrical coordinates. V 16 - x2 16 - x2 - y2 x² + y? dz dy…

A:

Q: 2. Evaluate the integral by first converting into spherical coordinates: a-y (a2-y2 a--y a2-x2-y2…

A:

Q: Evaluate the surface integral. J[x (x + y + z) ds, S is the parallelogram with parametric equations…

A:

Q: Let C be the closed, piecewise smooth curve formed by traveling in a straight line between the…

A:

Q: (x + y + z) ds, S is the parallelogram with parametric equations x = u + v, y = U-v, z = 1+2u+v, 0 ≤…

A:

Q: Use Stokes' Theorem to evaluate // (V x F) · dS where M is the hemisphere x? + y + z? = 9, x > 0,…

A:

Q: Evaluate the line integral / (6x + 2y)dx + (x – 7y)dy along the T curve C : x = 10 cos t, y = 20 sin…

A:

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

A: Use spherical coordinates to evaluate the triple integral SSSE x2 + y2 + z2 dV, Where E is the ball:…