valuate the line integral Scy da +a dy where C is the parameterized path a = t2, y = t°, 1

Q: Given The boundary of C: 0 ≤ x ≤ π,0 ≤ y ≤ sinx With Green's Theorem, Evaluate this line integral. $…

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Q: Evaluate the line integral -y dx + z dy + 2x dz, where C is the helix by 7(t) = (cos t, sin t, t),…

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Q: Evaluate the line integral ry dz for C : r(t) = (e , et+1)", te'), 0 <t<1 and then repeat with -C in…

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

Q: Compute the line integral (х — y) dx + xe x dy, where C is the closed curve consisting of two line…

A: line integration

Q: valuate the line integral, where C is the given curve. z2 dx + x2 dy + y2 dz, C is the line segment…

A: Follow next step

Q: Calculate the integral of f(z,y, 2) = 6z2 + 6y² + 2³ over the curve c(t) = (cost, sin t,t) for 0<t<…

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Q: Evaluate the line integral [x(1 с where C is given by the line r(t) = (1 + 3t, 2t), 0≤ t ≤ 1. x(1 +…

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Q: #6. Evaluate the line integral (y dx +z dy +x dz), where C is given by x = Vt, y = t, z = t? with…

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Q: (b) 6xy dV, where E lies under the plane z=1+x+y and above the region in the xy-plane bounded by the…

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

Q: 5xy dv, where E lies under the plane z = 1 + x + y and above the region in the xy-plane bounded by…

A: The given integral is ∭E5xydV. The region E lies under the plane z=1+x+y and above the region in the…

Q: Evaluate the triple integral (x + 2y) dV where E is bounded by the parabolic cylinder y = 7x² and…

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Q: Use a scalar line integral to find the length of the curve r(t) = (20 sin , 20 cos ,5), for 0 < t <…

A: Given

Q: Evaluate the line integral fe y dx + x dy where C is the parameterized path x = , y = f, 3 <iS9.…

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Q: Evaluate the line integral (xy dx + y dy), where C is the curve y = sin(x), 0 ≤ x ≤ π/2

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Q: Evaluate the integral |xy dx – z dy + 8z dz where C is the curve with parametric equations x = 2t, y…

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Q: 31 Evalvate the line integral gy lx + 8z dy tx dz, r(t) = (2+ t +3 for Ja9ydxt8zdyt xdz =

A: The line integral ∫C9ydx+8zdy+xdz, r(t)=2+t-1,t3,t2 for 0≤t≤1. We have to evaluate this integral.

Q: Evaluate the line integral S, y ds where C:x = t²,y= 2t, 0 <t < 3

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Q: Evaluate the line integral ex dx - xy dy +y dz, where c(t) = (1, t, t ), 0 sts 1. %3D

A: As per our guideline we are supposed to answer only first asked question.Kindly repost other…

Q: Evaluate xy dV where E is bounded by the parabolic cylinders y = x², x = y² E and the planes z = 0…

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Q: The line integral 2xe" dæ + a²e" dy + (1 – 2²)e¬²² dz is not independent of path.

A: The line integral ∫CF·dr is independent of path if the curl of F is zero, that is, ∇×F=0. Here, The…

Q: Evaluate the line integral, where C is the given curve. (x + 5y) dx + x2 dy y (5, 1) (6, 0)

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Q: Use Green's Theorem to evaluate the line integral Jo (y - x)dx + (2x - y)dy for the given path. C: x…

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Q: Evaluate the line integral ∫Cx2ds, where C is a the curve given by the equation y=f(x)=lnx, 1≤x≤e.

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Q: Evaluate the line integral Scy dx + x dy where C is the parameterized path x = t°, y = t³, 1 < t< 3.…

A: follow next step

Q: Evaluate the line integral cy dx + x dy where C is the parameterized path x = y = t° , 2<t< 6. Scydx…

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Q: (b) || 6xy dV, where E lies under the plane z 1+x + y and above the region in the xy-plane bounded…

A: A triple integral involves the integral with respect to x,y, and z. In this problem, we have to…

Q: The line integral x dy – y dx Jc x2 + y? is independent of path.

A: Given integral is ∫Cxdy-ydxx2+y2.

Q: Evaluate the line integral 2xyz dx + x²z dy + x?y dz where C is an oriented simple curve connecting…

A: Given line integral is,

Q: |[-y dx + xy dy]

A: See the attachment

Q: Evaluate the line integral x° + y) ds C Where C is given by r(t) = 3ti+t°j for 0<t<1

A: The region C is given by: r(t)=3t i + t3 j  ;       for  0≤t≤1

Q: Evaluate the line integral, where C is the given curve. z2 dx + x2 dy + y? dz, C is the line segment…

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Q: Calculate the line integral ∫Cyxdx+dy over the curve y=lnx in the interval 1≤x≤e (Figure 4).

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Q: . The uejveciion of a beam is governed by the equation d'y + 81y = 0 (x) dx %3D where o(x) is given…

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Q: Evaluate the line integral (4х + 5у)dx + (За — 6у)dy along the - curve C : x = 10 cos t, y = 20 sin…

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Q: Evaluate the line integral ∮C x dy−y dx where C is a circular, closed path defined by x^2+y^2=16

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Q: Evaluate the line integral. y dx + z dy + x dz, r(t) = (2 + t-1, t3, t2) for 0 <t< 3

A: Here, x=2+t-1 y=t3 z=t2

Q: Evaluate the line integral. y dx + z dy + x dz, r(t) = (2 + t-1, t³, t2) for 0 <ts 3

A: Given: From the above function, the following can be written,

Q: The value of the line integral S(2xy²dx + 2a?ydy + dz) along a path joining the origin and the point…

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Q: 2x dV, where E is bounded by the paraboloid x = 2y2 + 2z2 and the plane x = 2. %3D

A: The given integral is ∭E2xdV. The region E is bounded by the paraboloid x=2y2+2z2 and the plane x=2.…

Q: valuate the line integral J xev% ds where C is the line segment from > (1, 1, 1). This yields:

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Q: The line integral (x+/y) ds , where C is the line segment joining the points (0, 0) and (1, 1) , can…

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Q: Evaluate the line integral 9x²yzds with respect to s along the 2 curve C: x=t, y = t², z = t³(0 ≤ t…

A: Given: ∫C9x2yzds x=t y=t2 z=23t3 0≤t≤1 To find: Value of the given integral.

Q: valuate the line integral, where C is the given curve.       Cxy ds,    C: x = t2,    y = 2t,    0…

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Q: Evaluate the line integral. e ds where C is the curve given by x=2t, y=3t, 0<t<2

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Q: Evaluate the line integral ſ. 3(x – y) ds along the path C:r(t) = ti + (2 - t)j, 0<t<1

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Q: (2x + y)dx + (x + 2y)dy

A: Given integral is:  where C is any curve that joins the point (1,1) with the (2,2).

Q: Use Green's Theorem to evaluate the line integral. e cos(2y) dx - 2e* sin(2y) dy C: x2 + y? = a?

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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:…

Q: Set up the triple integral 8y dV, where E is bounded by the coordinate planes and the plane x + y +…

A: To Set up the triple integral for 8y dV, where E is bounded by the coordinate planes and the plane x…

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: