Use Stokes' theorem to evaluate (V x F) - dS where F(x, y, z) = -6yzi + 6rzj + 5(x2 + y²)zk and S is the part of the paraboloid z = x2 + y? that lies inside the cylino 2+ y? = 1, oriented upward.

Q: Evaluate the surface integral. xz ds, S is the part of the plane 2x + 2y + z = 4 that lies in the…

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Q: Calculate F· dS where F = (4x*z, 4y°z, 3:*) S is the surface of the solid bounded by the hemispheres…

A: Given

Q: Use Stokes' Theorem to evaluate fr. F(x, y, z) = xy i+x6j+z² k where Cis the intersection of the…

A: We can solve this using formula of Stoke theorem

Q: Evaluate the surface integral. || x²yz ds, s is the part of the plane z = 1 + 2x + 3y that lies…

A: To find ∬Sx2yz dS Here 0≤x≤5  and 0≤y≤2 S→=xi→+ yj→+…

Q: Evaluate the surface integral (2ayi – yzj + zk) · dS. Where S is the part of the paraboloid z = a2 -…

A: We have to evaluate the integral ∬S2xyi^−yzj^+zxk^⋅dS where S is the path of the paraboloid…

Q: Use Stokes' theorem to evaluate (ry – y) i – In zj+ (r+r²) k) - dr, where C is the intersection…

A: Let F be a vector field and C be the boundary of the surface S. The Stoke's theorem states that…

Q: S is the part of the paraboloid z = x² +y? below the plane z = 4 oriented upward and F(x, y, z) :…

A:

Q: Use Stokes' Theorem to evaluate fr F. dr. F(x, y, z) = xy i+ xºj + z² k where C' is the intersection…

A: To find- Use Stokes' Theorem to evaluate ∮CF·dr. Fx, y, z = xy i + x6 j + z2 k where C is the…

Q: Use cylindrical coordinates to evaluate the triple integral E Va2 + y? dV, where E is the solic 16…

A: use cylindrical coordinates of find the volume

Q: Evaluate the surface integral. x² + y2 + z• ds S is the part of the cylinder x2 + y2 = 25 that lies…

A: Given, Where, S is the part of the cylinder x2 + y2 = 25 which lies between the planes z = 0 and z =…

Q: |Use cylindrical coordinates to evaluate the triple integral |/L Vz? + y² dV, E is the solid bounded…

A:

Q: Evaluate the surface integral. /| xz ds, S is the part of the plane 2x + 2y + z= 4 that lies in the…

A: We need to find surface integral.

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

A:

Q: Evaluate the surface integral. J[x²³x² x²yz ds, S is the part of the plane z = 1 + 2x + 3y that lies…

A:

Q: Use Stokes' Theorem to evaluate F. dr where F(x, y, 2)æzî + 2ry} +3ryk where C is the boundary of…

A: According to the given information, it is required to use stokes theorem to evaluate the integral.

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: Use Stokes' Theorem to evaluate F. dr where F(x, y, z) = xi + yj +Xx² + y² )k and C is the boundary…

A: Stokes theorem: ∮f→·dr→=∬S curl f→·n^ ds Given F(x, y, z)=xi + yj+9(x2+y2)k and C is the boundary of…

Q: Use Stokes' Theorem to evaluate SSs curlF · dS, where F(x, y, z) = 7xyi + 3yzj+4z²k and S is the…

A: Consider the provided question, Use Stokes' Theorem to evaluate ∫∫Scurl F·dS,where F(x, y, z)…

Q: Calculate F.dS where F = (3ry", re", z*) S is the surface of the solid bounded by the cylinder y + z…

A:

Q: Let F = yz, xz, xy > . // curlF dS, where Use Stokes' Theorem to evaluate - x² – y? that lies above…

A: Given that F=-yz,xz,xy The paraboloid is z=8-x2-y2 and z=7 Finding the plane of intersection,…

Q: (b) Verify Stokes' theorem on a region which is a portion of the sphere r2 +y² +2 16 which lies…

A:

Q: Use cylindrical coordinates to evaluate the triple integral |||| Væ² +y² dV, where E is the solid…

A: Mentioned below is the answer

Q: Calculate F. dS where F = (4x°z, 4y*z, 32*) S is the surface of the solid bounded by the hemispheres…

A: Given that F→=(4x3z, 4y3z,3z4) and S is bounded by z=16-x2-y2, z=1-x2-y2, z=0 Now,…

Q: Use Stokes' Theorem to evaluate || curl F. dS. F(x, y, z) = x² sin(z)i + y²j + xyk, S is the part of…

A: Here use the stokes theorem to find the integral of the function and given that the function…

Q: Let o be the portion of the paraboloid F(u, v) = u cos(v)i+u sin(v)j+ u*k for which 0 < u< sin(v), 0…

A:

Q: Use Stokes's Theorem to calculate the surface integral / Ss F · dS where F(x, y, z) = xi +(2ze* –…

A: Given the vector field Fx,y,z=x,2zex-y,y. And S is a part of the ellipsoid 4x2+4y2+z2=17 between z=1…

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

A:

Q: Use Stokes' Theorem to evaluate F. dr, where F(x, y, z) = zi + yj – rk and C is the boundary of the…

A:

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

A: Given: F=-yz,xz,xy To Find: Use Stokes' Theorem to evaluate ∬Scurl F·dS, whereS is the part of the…

Q: Use Stokes' theorem to evaluate E (wy – y) i – In z j+ (x+x²) k) · dr, where C is the intersection…

A: The required curve C(red) is shown as follows: The intersection gives a disc D bounded by curve C,…

Q: .† Use Stokes' theorem to show that Sc(y+2)dx+ (z +x)dy+ (x+y)dz = 0, where C is the curve of…

A:

Q: Use Stokes' theorem to evaluate curl(F) · dS. F(x, y, z) = x² sin(z) i + yj + xyk, S is the part of…

A: Given that, And he paraboliod is The aim is to find The radius is 3. Take C is x(t)=cos t and…

Q: Use Stokes' Theorem to evaluate [[(curlF)•ndS where F(x,y,z) =x’i+z°j-y°k and o is that portion of…

A:

Q: Use Stokes' theorem to evaluate $ (xy – y) i – In z j+ (x +x²) k) • dr, where C is the intersection…

A:

Q: 5. Use Stokes theorem to evaluate J[V x F-ds over the surface of the paraboloid z = 5-x² - y² above…

A: If S  be   smooth surface bounded by a  closed , positively  oriented  smooth boundary curve C with…

Q: Use Stokes' Theorem to evaluate F- dr. F(x, y, z) = xyi + x*j+z?k where Cis the intersection of the…

A:

Q: A parametrization of the portion of a surface z=x²+y2 below the plane z= 16 is given by 7(u,v)=…

A: Since you have sent more than one question so we can just answer one of them because of our…

Q: '. Use Stokes' Theorem to compute || CurlF •dS for the hemisphere x + y +z² =9, z20, intersected…

A:

Q: Integrate g(x,y,z) = x + 5y + z over the portion of the plane 4x + 8y + z = 8 that lies in the first…

A:

Q: Find the surface area of the plane 3x + 2 y + z = 6 that lies in the first octant.

A: The surface area of the plane z=f(x,y) is given by the formula                          S=∬D…

Q: 2. When F = (y z, xz, x² y?) and S is the part of the paraboloid z = x? + y? that lies in the…

A: First, we have to parametrize the surface to apply the Stokes theorem.  Part of the paraboloid is…

Q: Calculate F. dS where F = (3xy°, xe", zª) S is the surface of the solid bounded by the cylinder y? +…

A: Let's find.

Q: Use Stokes' Theorem to evaluate curl F· dS. F(x, y, z) = x² sin(z)i + y²j + xyk, S is the part of…

A: Given problem is :

Q: 2. Use Stokes' Theorem to evaluate curlF.dS whereF = 2y i + 3z j+(z – z) k and S is the portion of y…

A:

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

A: Strokes theorem:

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: Evaluate the surface integral. SS, xz ds, S is the part of the plane 2x + 2y + z = 4 which is in the…

A:

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: Find the centroid of the solid bounded above by the paraboloid z = x? + y?, below by the plane z = 0…

A: Let S be the solid region bounded by (i)   Paraboloid  z=x2+y2 from above (ii)  Plane z=0 from below…

Q: Let F = · Use Stokes' Theorem to evaluate curi curlF · d§, where S is the part of the paraboloid z…

A: here i had written stokes theorem and done some basic calculation to find the curve of intersection…

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: