Evaluate the line integral , F · dr,  where C is given by the vector function r(t). F(x, y, z) = sin(x) i + cos(y) j + xz k   r(t) = t3 i − t4 j + t k,  0 ≤ t ≤ 1

Q: Evaluate the line integral using Green's Theorem and check the answer by evaluating it directly.…

A: Here , we have given an integral and we have to integrate over a rectangle whose vertices are given…

Q: Evaluate the integral f. x'y°dy – x³y?dx, where C is the square determined by the 0, y = 0, x =…

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Q: Evaluate the line integral F dr, where C is given by the vector function r(t). F(x, y, z) = sin(x) i…

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Q: Use Green's Theorem to evaluate the line integral below. | (2 - v?) dx + 5xy dy C: r = 1 + cos(0), o…

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Q: Evaluate the line integral Sc F dr where F(x, y, z) = (x+y)i+ (y – 2)j+z²k and r(t) = t²i+t³j+t?k, 0…

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Q: Evaluate the line integral f,xy dx + x² dy, where C is given by y = x³,1 < x < 2. 33 132 31 12 5 124

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Q: Evaluate the line integral, where C is the given curve. Sc (2x + 9z)ds C: x = t, y = t2, z = t 3, 0…

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Q: Evaluate the line integral, where C is the given curve. z dx + x dy + y dz, C: x = t5, y = t, z = t,…

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Q: Calculate the line integral [(of).do where f(x, y, z) = x ² ¹² + 4 xy² z yz с + е хуча and C is the…

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Q: Evaluate the line integral along the curve C. (xz + y) ds, C is the curve r(t) = (-4 - t) i + 2t j -…

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Q: Calculate the line integral of h (x, y) = (2xy) i+ (x² – 2ry) j; over r (1u1) = cos u i+ sin u j, u…

A: Here,hx,y=2xy i + x2-2xy jr(u)=cosu i+sinu j

Q: Evaluate the line integral f. 3x?yz ds along C:x = t,y = t²,z = ÷t³ , 0 <t<1

A: Use the following formulae, to obtain the solution.…

Q: Compute the line integral of the scalar function f(x, y, z) = 2x² + 8z over the curve c(t) = (e',…

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Q: Evaluate the line integral ſ F. dr, where F(x, y, z) = sin x i + cos y j + xz k, where C is given by…

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Q: Evaluate the line integral ∫cF⋅dr, where F(x,y,z)=−3sinxi−2cosyj−xzk and C is given by the vector…

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Q: Evaluate the integral f. x'y°dy – x³y'dx, where C is the square determined by the lines x = 0, y =…

A: We have to evaluate the given integral.

Q: Let o (x +e*siny) dx + (x +e* cos y)dy be the line integral on the boundary of the triangle enclosed…

A: We will find out the required value.

Q: Evaluate the line integral along the curve C. 1. 21 10 21 2. 20 21 3. 20 8 ds, C is the curve r(t) =…

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Q: B) Evaluate the line integral of the function P(x,y) dx + Q(x,y) dy. over the curve C where…

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Q: Find the work done by force F over the given paths in the direction of increasing t. F = zi + xj +…

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Q: Suppose that f(x, y) = xy and that C is the curve 7 (t) = (t², t³), where 0 <t <1. Fill in the…

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

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Q: Given r(t)=t i+2t j and 0 <t< 1. Evaluate the line integral f,(x+y) ds = fr(1)r'(1) dt. a

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Q: Let p (x +e*siny) dx + (x +e* cos y)dy be the line integral on the boundary of the triangle enclosed…

A: To find the equivalent double integral we will use the Green's Theorem also we will draw the…

Q: Evaluate the given integral by using a transformation to cylindrical coordinates: f (x2+2)¹/2 dv,…

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

Q: Evaluate the line integral F dr, where C is given by the vector function r(t). F(x, y, z) = sin(x) i…

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Q: 9. Evaluate the line integral of F = (2xy + 1,x² + 2yz, y? – 2) over the region C parametrized by…

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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 F· dr, where C is given by the vector function r(t). F(x, у, z) sin(x) i…

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Q: Evaluate the line integral F. dr, where F (x, y) = (3+2xy) i+ (x² – 3y²) j and C is the curve given…

A: Given problem:-

Q: Calculate the line integral of h (x, y) = (2xy) i + (x² – 2xy) j; over r (u) = cos u i+ sin u j, u e…

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Q: 3. Evaluate the line integral c F. dr where F(x, y, z) = (e², xz, (x + y)) with 0 ≤ t ≤1 over the…

A: The solution is given as

Q: Compute the line integral of the scalar function f(z,y, 2) = 2z² + 8z over the curve e(t) = (e*, t²,…

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Q: Evaluate the line integral using Green's Theorem and check the answer by evaluating it directly. p 2…

A: Given problem:-

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: c2π rieit 0 20 + reit dt

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Q: Evaluate the line integral INTEGRAL C (x − y 2 )ds where C is given by r(t) = (t^2 + 1)i + (t +…

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

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Q: Evaluate the line integral F. dr, where C is given by the vector function r(t). F(x, y, z) = sin(x)i…

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Q: Evaluate the line integral / F. dr, where C is given by the vector function r(t). F(x, y, z) = xi +…

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Q: Compute the line integral of the scalar function f(x, y, z) = 2x² + 8z ver the curve c(t) = (et, t2…

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Q: Compute the line integral of the scalar function ƒ(x, y, z) = 2x² + 8z over the curve c(t) = (e¹,…

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Q: Consider the following line integral. xy dx + x?y dy, Cis counterclockwise around the triangle with…

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Q: Let F = (x + 2xy,x² – y). a. Given that F is conservative, find the potential function ø where F =…

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Q: Let f(x,y,z)=x^(2)y+y^(2)z.  Find the exact value using the Fundamental Theorem of Line Integrals,…

A: Given That :   f(x,y)=x2y+y2zr(t)=sinπ2t ,3t+log(2-t),ecos(πt) a=0,b=1 To find : exact value using…

Q: 9. Evaluate the line integral ſo F · dr, where F(x, y) = (ry, 3y?), and C is is given by the vector…

A: We have to integrate F • dr over the curve C given by the vector function r(t) = ( t4, t3 ) for t in…

Q: Find the line integral of F(x, y, z) = x'i + yj+ zk along the line segment C from the origin to the…

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Q: Let F(x, y) = 2xy³i+ (1+ 3x?y?)j. (a) Demonstrate that F is conservative. (b) Find the potential…

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Q: Find the integral of ſ r(t) dt for the vector-valued function, r(t) (In(t))i + /1+ cos4t)j + dt k…

A: NOTE: The vector-value function is dependent on variable t, so z component of vector function is…

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: