1. Use Stokes' Theorem to evaluate curlF.dS where F =z'i+ (4y-y) j+2zk and S is the portion of z = r + y -3 below z = 1 with orientation in the negative z-axis direction.

Q: Change the Cartesian integral to an. equivalent polar integral e -(x²+y²) dy dɔ JT - T/2 re-* dr dt…

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Q: Find the work done by the force F(x,y, z) = x*j + 5zk over the curve C: r(t) = cos t i+ sintj+ tk,…

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Q: Let F = %3| Use Stokes' Theorem to evaluate curlF · dS , where S is the hemisphere x? + y? + z² =…

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Q: 4. Use Stokes’ theorem to evaluate fa F. dr, where F(x,y, z) = (xy,yz, zx), and C is the triangle…

A: I am going to solve the given problem by using some simple calculus to get the required result.

Q: Verify Stokes' theorem for F = (x+y-4) i+3xy j+ )k, over the surface of the hemisphere x2 + y2 + z2…

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Q: Let F = Use Stokes' Theorem to evaluate F dī, where Cis the curve of intersection of the parabolic…

A: Given F¯=xy,4z,6y We have to evaluate ∫CF¯⋅dr¯ by using stokes theorem, where C is the curve of…

Q: Find the area under the parametric curve x=t(1+t) , y=1/(1+t) ,  between t= [0,1]

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Q: Evaluate the line integral / V(ry + xz + yz) · dĩ over the curve F(t) = (t, 2t, 3t) for 0 <t < 4…

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Q: Use Stokes' theorem to evaluate curl(F) · dS. F(x, y, z) = exy cos(z)i + x2zj + xy k, S is the…

A: F(x, y, z) = exy cos (z) i + x2 z j + xy k

Q: Consider the surface S shown in the gure and whose parameterization is given by: r(u, v) = (sin v,…

A: Consider the surface S shown in the gure and whose parameterization isgiven by: r(u, v) = (sin v, u…

Q: Use Stokes' Theorem to compute 1 z? dx + (ry) dy + 2020 dz, where C is the triangle with vertices at…

A: Given I=∮C12x2+xydy+2020dz where C is the triangle with vertices at 1,0,0 , 0,2,0 , 0,0,2 traversed…

Q: Give the integral representing the length of the parametric curve x=t^3, y=t^4, 0≤t≤1.

A: Given: A parametric curve:   Now the length of the parametric curve is given by the formula:

Q: Find the coordinates of the centroid of the curve x = cos t, y = t + sin t, 0&lt;=t &lt;= pai.

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Q: Compute (VxF). dS, where S is given by x² + y² + (z − 1)² = 2 and z ≥ 0, F(x, y, z) = (-esinzy, x +…

A: The general form of a vector-valued function is given by F1,F2,F3 where F1,F2 and F3 are the scalar…

Q: Evaluate the line integral I of the one-form w = xy dx + 12y dy over the parametric curve a : [0,…

A: Given: ω=xydx+12ydy To find: I=∫αω  where α:[0,ln2]→ℝ2 with α(t)=(e2t,e-2t) ∴x=e2t,y=e-2t Now,…

Q: CurlF dS x+y' +z =9, z20 se Stokes' Theorem to compute s for the hemisphere intersected with the xy-…

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Q: 18 Verify Stokes' theorem F.dr = (curl F)· dS C S where F = (x² + y – 4, 3xy, 2xz +z?) and S is the…

A: We have

Q: Consider the curve described by the parametric equations x = t° +1, y= t² – º. Compute the area…

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Q: 339. Use Stokes' theorem to evaluate curl F · dS, where F(x, y, z) = e®Ycos zi + æ²zj+ xyk, and S is…

A: please see the next step for solution

Q: 3. Consider A(r) = (y, -x,0), and the line integral A• dr along the C boundary of the plane x + y +…

A: The given vector function is A(r)=(y,-x,0)., the line integral ∮CA·dr is along the boundary of the…

Q: Use Stokes' Theorem to evaluate curl F· dS. F(x, y, z) = zeYi + x cos(y)j + xz sin(y)k, S is the…

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Q: Let F = . Use Stokes' Theorem to evaluate / curlF · d5, where S is the hemisphere x² + y² + z² =…

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Q: Evaluate the unoriented line integral (2æ² + y² + z²) ds along the parametric curve a : [0, 1] → R³…

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Q: Use polar coordinates to evaluate the double integral V1 1-z2 dy dæ. (1+z² + y?)²

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Q: S,F•dr, and C is the triangle, with Use Stokes Theorem to evaluate where F(x,y; =) = (x+ y² )i+ (vy+…

A: Given: Fx,y,z=(x+y2)i+y+z2j+z+x2k and C is the triangle with vertices 1,0,0 ,0,1,0 and 0,0,1 We know…

Q: 11. Let F = (-xy + xz²)i + (4xz + y²)j + e² k. Use Stokes' Theorem to calculate fcF. dr, where C is…

A: We have to find∫cF→.dr→.

Q: 1. Compute ff,(V × F) - dS, where S is given by r² + y? + (z – 1)? = 2 and z > 0, and F(x, y, z) =…

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Q: 1. Compute ff(V × F) - dS, where S is given by r² + y²+ (z – 1)² = 2 and z > 0, and F(r, Y, z) =…

A: Given: S is given by x2+y2+z-12=2  and z≥0 Fx,y,z=-esinzy,x+sinz,cosx

Q: If S c R³ is the unit sphere and F Sls curl F n dS. (e*y, sin(22 + y), xyz), then evaluate the…

A: We will find out the required value.

Q: Use Stokes' theorem frotF. ds to evaluate F(x, y, z) = (x + tan¯¹yz)i + y²zj + zk, S is the part of…

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Q: Use Stokes' Theorem to evaluate L* + cos x) dx + (sin y + z²) dy + x dz where C is the closed curve…

A: To Evaluate: Use Stoke's Theorem:

Q: Let F < ryz, ry, r'yz Use Stokes' Theorem to evaluate curlF dS, where S consists of the top and the…

A: The given surface is a cube, where the bottom face is not considered. The surface in question has 4…

Q: - Prove Stokes' Theorem by verifying it for F= [y,z,x] and S the paraboloid. %3D te CEY2=(x, y) = 1…

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Q: curl F dS, where Use Stokes' Theorem to determine which of the following is equivalent to F(x, y, z)…

A: The given function is F(x, y, z) = 2x2 i^ + xy j^ + ez k^.The surface S is the part of the…

Q: Let F . F. dr, where Use Stokes' Theorem to evaluate C is the curve of intersection of the…

A: The given vector function: F→=&lt;xy,2z,3y&gt; To find ∫CF→.dr by using Stokes theorem. The Formula…

Q: Let F = Use Stokes' Theorem to evaluate F. dī, where C is the curve of intersection of the…

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Q: 3. Suppose F z = 4 – a? – y? above the r- y plane, oriented upward. Calculate fe F dĩ in two ways:…

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Q: Let S be the surface parametrized by x = r cos 0 y = r sin 0 z = 30 where and 0<r< 4 Evaluate the…

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Q: The motion of a fluid particle is defined parametrically as follows, r(t) = sinh(t), and y(t) =…

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Q: Given F(x, y, z) = -yzį + xzj – xyzk be a vector function that passes through a surface S of z +x? +…

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Q: Use Stokes's Theorem to show that 0= [[ curl(F) - dS where F(r, Y, z) = (2, 2ry, x + y) and S is the…

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Q: Use polar coordinates to evaluate the double integral. 1 1 + x² + y² where R is the sector in the…

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Q: Compute the line integral | (yze*v= + y° + z - 2x) dr + (xze*# + 2ry) dy + (rye#"= + x + 32²) dz,…

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

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Q: Evaluate the line integral ſp(xi + Vyi) · dr where D is the oriented curve shown below. (1,1) y = x2…

A: We know that r=xi+yj As, the path followed is y=x2 So, r=xi+x2j Let x=t, then  r=ti+t2jdrdt=i+2tj…

Q: 2). Use Stokes' theorem to evaluate the integral 6 F • dř for F(x, y, z) = (x – y)î + (y – z)ĵ + (z…

A: Formula: By Stoke's theorem we have ∮C F · dr¯  =∫t=t1t=t2 F¯(r(t))r'(t) dt, where r(t) is…

Q: Define stokes theorem and evaluate the line integral SF.r'(s) ds where F = [4z, 2x, 2x], C: the…

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Q: Use Stokes' Theorem to evaluate curl F. dS. F(x, y, z) = exY cos(z)i + x²zj + xyk, S is the…

A: Given F→x,y,z=exycoszi^+x2zj^+xyk^ We have t find the integral ∬ScurlF→⋅dS, where S is the…

Q: Use Stokes' Theorem to evaluate |/. CurlF dS, where F(r, y, 2) = xyzi + xj + e" cos zk and S is the…

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Q: 'Use Stokes' Theorem to compute | 1 F. dr where C is the curve where the cylinder a? + y? = 1…

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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: