Evaluate the line integral (cos x + sin y) ds; C is the line segment from (0, 0) to (x, 67).

Q: Evaluate the line integral / Vxy° +3 dx +6xy" dy, where C any curve joining (1, -1) to (16, 1).

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Q: Evaluate the line integral, where C is the given curve. | xyz? ds, C is the line segment from (-1,…

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Q: Evaluate the line integral along the path C given by x = 2t, y = 8t, where U + dy

A: Given

Q: Evaluate the line integral along the path C given by x = 4t, y = 8t, where 0 ≤ t ≤ 1. [(x + 3y²) dy…

A: Gievn, x=4t,y=8t And 0≤t≤1

Q: Evaluate the line integral along the path C given by x = 2t, y = 6t, where 0 s t s 1. (y - x) dx +…

A:

Q: Evaluate the line integral f e-*(cos y dx - sin y dy), where c is the rectangle with vertices (0,0),…

A: The line integral function is ∫Ce-xcosydx-sinydy. The vertices of the rectangle is 0, 0, π, 0, π,…

Q: Evaluate the line integral, where C is the given curve. | xyz? ds, C is the line segment from (-2,…

A: Find the equation of line in parametric form.

Q: Evaluate the line integral along the path C given by x = 2t, y = 4t, where 0 ≤t≤ 1. √(x + 3y²) dy…

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Q: Evaluate the line integral along the path C given by x = 2t, y = 6t, where 0 s ts 1. %3D (y – x) dx…

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Q: Evaluate the line integral, where C is the given curve. (X/y) ds, C: x = t3, y = t,

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Q: Evaluate the line integral. y dx - x dy, parabola y = x² for 0 < x < 8

A: Given:

Q: Evaluate the line integral, where C is the given curve. [x² x sin(y) ds, C is the line segment from…

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Q: Evaluate the line integral Lav where C is given by the line y = lnx, 1 ≤ x ≤e. xy ds,

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Q: Evaluate the line integral along the path C given by x = 2t, y = 4t, where 0 sts 1. (y - x) dx +…

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Q: Show that the integral is independent of the path, and use the Fundamental Theorem of Line Integrals…

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Q: Evaluate the line integral, where C is the given curve. |x sin(y) ds, C is the line segment from (0,…

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Q: Evaluate the integral (+ 2) dz where C is the line segment from the origin to the point 1+ 2i.

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Q: State two other forms for the line integral ∫C F ⋅ n ds given that F = ⟨ƒ, g, h⟩ .

A:

Q: Evaluate the line integral along the path C given by x = 4t, y = 8t, where 0 st< 1. + 3у?) dy

A: We need to evaluate the following line integral ∫Cx+3y2dy Where, C is defined as x = 4t, y = 8t ;…

Q: Evaluate the line integral, where C is the given curve. y° ds, C: x = t°, y = t, osts 3

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Q: Find the line integral of ƒ(x, y) = x - y + 3 along the curver(t) = (cos t)i + (sin t)j, 0 &lt;_ t…

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Q: Calculate the double integral. 9x sin(k + y) dA, R = 0, 풍핑x [o, 플]

A: First we will evaluate the inner integral wrt 'y' and then evaluate wrt 'x'.

Q: Evaluate the line integral, where C is the given curve. | xyz? ds, C is the line segment from (-2,…

A:

Q: Evaluate the line integral, where C is the given curve. (x + 9y) dx + x² dy, C consists of line…

A: To evaluate the line integral, consider the curveC1 consisting of the line segment from0,0to9,1. The…

Q: Evaluate the line integral Say dx +z dy, where C is the line segment from (-1, 1, 2) to (2, 3, 1).

A: Evaluate the line integral ydx +zdy where C is the line segment from (-1,1,2) to (2,3,1). Here i…

Q: Evaluate the line integral ,F. dr where F = (sin x, –2 cos y, 5xz) and C is the path given %3D by…

A: Integral F.dr = integral F(r(t)).r'(t) dt

Q: Evaluate the line integral along the path C given by x = 2t, y = 6t, where 0 sts 1. dx + 5x2y2 dy

A:

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: Calculate the double integral. 7x sin(x + y) da, R = [0, * [0, =]

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Q: Evaluate the line integral (e*y+ 2x) dx + (e* + 1) dy, where C is a smooth curve from the point (0,…

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Q: Find the line integral of F =9xy*ï +(x² + 3),yj over the path L.

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Q: Evaluate the line integral ds along the curve C given by 7()=(2 sint, 2 cost), 0sts T.

A: Given line integral is ∫Cx+12yds. Curve is r⇀t=2sint,2cost,0≤t≤π. xt=2sint,yt=2cost Compute the…

Q: Show that the integral is independent of the path, and use the Fundamental Theorem of Line Integrals…

A: Line integral will be path independent if vector field whose integral we need  is conservative…

Q: Evaluate the line integral - y² dx, where C is the square cut from the first quadrant by the lines x…

A:

Q: Evaluate the line integral (2 - u) dz + (3 + æ) dy] where C is the parabolic arc = a² from (0, 0) to…

A: We have to evaluate the line integral.

Q: Evaluate the line integral, where C is the given curve. | xyz? ds, C is the line segment from (-3,…

A: Integration

Q: Evaluate the line integral, where C is the given curve. ∫(with c in subscript) x sin y ds, C is the…

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Q: Calculate the integral | (2x + sin y) dx + x cos y dy 7, where L is any curve from the A(0,0) to B (…

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Q: Evaluate the line integral along the path C given by x = 2t, y = 8t, where 0 sts 1.

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Q: Find the line integral of ƒ(x, y) = x - y + 3 along the curve r(t) = (cos t)i + (sin t)j, 0&lt;=…

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Q: Evaluate the line integral, where C is the given curve. |(2y3 - Vx)dy, c is the arc of the curve y =…

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Q: Evaluate the integral a sin y ds where C is the line segment from (0, 3) to (4,6),

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

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Q: Evaluate the line integral along the path C given by x = 2t, y = 8t, where 0 sts 1. 3у?) dy

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Q: Evaluate the line integral along the path C given by x = 2t, y = 4t, where 0 stsi. (y - x) dx +…

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Q: Find the line integral along the given path C. Ju- |(x- y)dx, where C: x = t, y = 5t + 6, for 0 sts2…

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Q: Evaluate the line integral, where C is the given curve. (4x + 18z) ds, C: x = t, y = t2, z t3, 0sts1

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Q: Evaluate the line integral along the path C given by x = 2t, y = 4t, where 0 <ts 1. X + dy

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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: Evaluate the line integral, where C is the given curve. x sin(y) ds, C is the line segment from (0,…

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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: