S is the part of the paraboloid z = x² +y? below the plane z = 4 oriented upward and F(x, y, z) : Evaluate Sf F · dS. (x² + (sin z)e³ sin(y²), xy, z).

Q: If E is the solid that lies under the plane z = 3 which its base on the xy-plane is between the two…

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Q: 23. Use Stokes' theorem to evaluate Se F · dr, where F(x, Y, z) = (xy, yz, zar) and C is the…

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Q: Use Stokes' theorem to evaluate (V x F) - dS where F(x, y, z) = -6yzi + 6rzj + 5(x2 + y²)zk and S is…

A: Given that F(x,y,z)=-6yzi^ + 6xz j^+ 5(x2+y2)z k^ Calculating the curl,…

Q: Calculate F · dS where F = (e* sin cos y, yz?) y, et and S is the surface of the box bounded by the…

A: By using the divergence theorem, we can make equality between the surface integral of a function and…

Q: y dV, where B is the portion of the unit ball B = {(x, y, z) : x² +y² + z² < 1} which lies below the…

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Q: I y dV, where G is the solid enclosed by the plane z = y, the xy-plane, and the parabolic cylinder y…

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Q: Evaluate fs F dS, where F(r, y, z) = (2², xy, z) and S is the part of the paraboloid z = 22 + y?…

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Q: Evaluate e dV where E is enclosed by the paraboloid z = 5+x +y, the cylinder a2 + y = 1, and the xy…

A: The objective is to evaluate ∫∫∫EezdV, where E is enclosed by the paraboloid z=5+x2+y2, the cylinder…

Q: e dV where E is enclosed by the paraboloid z =1+ a? + y", the cylinder a? + y = 3, E Evaluate and…

A: We will use triple integration here

Q: Use cylindrical coordinates to valuate fff, z dV where E is enclosed by the paraboloid z = x² + y?…

A: To evaluate ∫∫∫Ez dV E is enclosed by the paraboloid z=x2+y2 and the plane z = 16 z is bounded by r2…

Q: Calculate F. dS where S F = (e" sin y, e* cos y, yz²) and S is the surface of the box bounded by the…

A: Given Data: Integral: ∬SF→·dS Vector: F→=exsiny, excosy, yz2 Planes: x=2, y=4, z=3 The divergence of…

Q: Find the flux of F (x, y, z) = 4.xi+ 4yj+ 2k outward (away from the z-axis) through the surface cut…

A: The field lines of a vector field F through surfaces with unit normal n, the angle from n to F is θ.…

Q: Evaluate e dV where E is enclosed by the paraboloid z = 5 + x² + y°, the cylinder 2² + y? = 5, and…

A:

Q: (b) f(x, y) dA where D lies below the paraboloid z = x2 + y?, above the ry-plane, and inside the %3D…

A: Let's find.

Q: Evaluate ez dV where E is enclosed by the paraboloid z = 4 + x? + y², the cylinder x? + y? = 1, and…

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Q: 2. Use Stokes theorem to evaluate fs da. a) a = x2 sin z da + y? dy + ry dz, S is the part of the…

A: As per our guidelines, we are supposed to answer only one question and rest can be reposted Given:…

Q: a) Evaluate (-ri-yj+2z k) -n dS, where o is the portion of the paraboloid z = 3r + 3y? between the…

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Q: Use cylindrical coordinates to evaluate SI vz dv where E is the solid E bounded by the cylinder,…

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Q: Evaluate fff (3x + 2y)dV, where S is the solid bounded above by the plane z=4, below by plane z=0…

A: ∫∫∫(3x + 2y) dV=∫∫∫(3x + 2y) dxdydz = ∫z = 04∫y = -13∫x =-11(3x + 2y) dxdydz = ∫z = 04∫y = -134y…

Q: Calculate F. dS where F = (e" sin y, e* cos y, yz“) S is the surface of the box bounded by the…

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Q: 5 + 2? + y, the cylinder Evaluate e dV where E is enclosed by the paraboloid z = E 2 + y? = 1, and…

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Q: 2. Use Stokes' Theorem to evaluate , Vx F ·n dS where F(x, y, z) = and C is the part of the sphere…

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Q: Find the surface area of the paraboloid z =1-x² - y? that is above the xy-plane.

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Q: Let F = . Use Stokes' Theorem to evaluate curlF · dS , where S is the part of the paraboloid z = 14…

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Q: Find the area of the part of the plane 3x + 3y + z = 9 that lies inside the cylinder x2 + y2 = 25.

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Q: Find // y dV, where E is the solid bounded by the parabolic cylinder z = x? and E the planes y O and…

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Q: Evaluate e dV where E is enclosed by the paraboloid z = 5 + x² + y°, the cylinder 2² + y? 1, and the…

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Q: V9/ Compute I ,cxyz), uhere s is the part f the plane +x that lies inside the half - cyllnder…

A: Given: The Integral is ∬SxyzdS. The surface is the part of the plane S  z=y+x that is inside the…

Q: Find the area of the surface. The part of the paraboloid z = 1 – x2 - y? that lies above the plane z…

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Q: Evaluate e dV where E is enclosed by the paraboloid z = 6 + x² + y°, the cylinder 2? + y? = 5, and…

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Q: Use Stokes' theorem to evaluate V ×F· î dS where = (9y²z, 4xz, 2x²y²) and S is the paraboloid z = x²…

A: Step:-1 Stoke's Theorem:- ∬s∇×F→.n dS=∫F.dr Now, given that F→=9y2z, 4xz, 2x2y2 and paraboloid…

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

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Q: Evaluate e dV where E is enclosed by the paraboloid z = 2 + x² + y², the cylinder æ? + y² = 5, and…

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Q: Evaluate e dV where E is enclosed by the paraboloid z = 4+x² + y°, the cylinder x² + y? = 3, and the…

A: Here we have to evaluate ∫∫∫E ez dV , where E is the enclosed by the paraboloid z=4+x2+y2 , the…

Q: 5) 2z dV, E is the solid enclosed by the cone % = Vr2 + y? and the E plane z 5

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Q: Let E denote the part of the plane x + y + z = 1 that is inside the cylinder r? + y? = 1. Compute…

A: We have to find the value of  ∬ F→.dS→

Q: Evaluate (xr + y² + =³)? dV, where 2 is the solid hemisphere (half of the Ω sphere) enclosed by x +y…

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Q: C is the intersection curve between the paraboloid z = x² + y? and the plane 6x + 8y – z = 21,…

A: Strokes theorem:

Q: Use Stokes' Theorem to evaluate ∫CF⋅dr where F(x,y,z)=xi+yj+3(x2+y2)k and C is the boundary of the…

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Q: The surface area of the part of the paraboloid z=x+y that lies under the plane z=2 is: Z=,

A: Surface area is given as A=∬D1+δzδx2+δzδy2dA

Q: The surface area of the part of z = 2+ /4 – x² – y? that lies inside the paraboloid z = x² + y? is

A: Given is the equation of a surface and a paraboloid. The objective is to find the surface area. The…

Q: Let F = . Use Stokes' Theorem to evaluate curlF · dS , where S S is the part of the paraboloid z = 9…

A: Find your answer below

Q: a) Evaluate (-x i - yj+ 2z k) - n dS, where o is the portion of the paraboloid z = 3r2 + 3y? between…

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Q: Find y dV, where E is the solid bounded by the parabolic cylinder z = a and the planes y = 0 and z =…

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Q: Find the circulation of F(x, y, z) = (-ysin z, x sin z, xy cos z + 12) around the circle cut from…

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Q: Evaluate e dV where E is enclosed by the paraboloid z = 5 + x + y°, the cylinder + y = 1, and the ry…

A: To Determine: evaluate ∫∫∫eZdV  where E is enclosed by the paraboloid z=5+x2+y2,the cylinder…

Q: yz ds where o is the part of the plane x + y+ z = 1 that lies in the first octant.

A: Given The given integral ∫∫σyzds over the region x+y+z=1 that lies in the first octant can evaluated…

Q: Find SS F dS where F(x, y, z) = z i+(3x – 1)j+ yk and S is the portion of the plane x + y + z = 1…

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Q: Use Stoke’s Theorem to evaluate the line integral ∮C(z2−y2)dx +(x2−z2)dy +(y2−x2)dz, where the curve…

A: Given equation of the curve is ∮C(z2-y2)dx+x2-z2dy+(y2-x2)dz ,the equation of the paraboloid…

Arc Length

Arc length can be thought of as the distance you would travel if you walked along the path of a curve. Arc length is used in a wide range of real applications. We might be interested in knowing how far a rocket travels if it is launched along a parabolic path. Alternatively, if a curve on a map represents a road, we might want to know how far we need to drive to get to our destination. The distance between two points along a curve is known as arc length. 

Line Integral

A line integral is one of the important topics that are discussed in the calculus syllabus. When we have a function that we want to integrate, and we evaluate the function alongside a curve, we define it as a line integral. Evaluation of a function along a curve is very important in mathematics. Usually, by a line integral, we compute the area of the function along the curve. This integral is also known as curvilinear, curve, or path integral in short. If line integrals are to be calculated in the complex plane, then the term contour integral can be used as well.     

Triple Integral

Examples: