Use Green's Theorem to find the integral rdy - dr where C is the curve consisting of three line segments: from (0, 0) to (4,0), next from (4,0) to (6, 2), and lastly from (6,2) to (0,0) (A) 0 (B) 4 (C) 8 (D) 12

Q: Show that the line integral is independent of path by finding a function f such that Vf = F. 2xe-Ydx…

A:

Q: Let Li be the straight line segment from (0.0,1) to (1,a, 8) and let F: (z+ax, ay, x> Calculate the…

A:

Q: Find the solution in the form of Fourier integrals: Ut- 3uxx = 0, u(0, t) = 0, |u(x, t) bounded as x…

A: As per the question we are given a initial boundary value problem as : ut - 3uxx = 0 And we have to…

Q: 3. Compute the scalar line integral 3 y – xz ds where C is the line seg- ment from (0, 1, 1) to…

A:

Q: Show that the following line integral is path independent and evaluate its value. Show the details…

A:

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

A:

Q: Evaluate the line integral, where C is the given cu curve. (x + yz) dx + 2x dy + xyz dz, C consists…

A: C consist line from A (2, 0, 1) to B(3, 3, 1) Now,

Q: Show that the line integral is independent of path by finding a function f such that Vf = F. 2xe-Ydx…

A:

Q: 1) Evaluate the line integral Fdr where F(x, y, =) = (x+3y, 2y +z, xy) and y: line segment from…

A: Here, F→(x,y,z)=x+3y,2y+z,xy and γ: line segment from (2,1,3) to (3,3,5) We need to evaluate ∫γF→dr→

Q: Verify the Green's Theorem to evaluate the line integral . (2y²dx + 3xdy),

A:

Q: Use Green's theorem in order to compute the line integral £ (x – 3)³ dy – y³ dx where is the circle…

A:

Q: 4. Use Green's Theorem to evaluate the line integral | 2xydx+ xy'dy, where C is the triangle with…

A:

Q: line integral is independent of path by finding a function f such that ∇f = F

A:

Q: Calculate line integral F-dr where F(x, y) = (7x + y, 5x + 2y) and C is a rectangle with vertices…

A:

Q: Evaluate the line integral of x^3(z)ds, where C is the line segment from (0,8,2) to (7,4,6).

A:

Q: P is a continuous path formed by connecting the following two lines segments: C: from (1,0,0) to…

A:

Q: Verify Green's Theorem by evaluating both integrals aN dA |v² dx + - x² dy ax ду for the given path.…

A:

Q: Evaluate the line integral ∫c x3 z ds, where C is the line segment from (0,8,8) to (5,7,6).

A:

Q: 2. Evaluate directly the line integral fo a dy where C is the quarter-circle from (0, 4) to (4, 0)…

A:

Q: Compute the integral of f(x) = 0.4 – 2x² + 40x5 from a = 0 to b = 0.8 using i. Trapezoidal Rule with…

A: Numerical integration

Q: . Compute the line integral of E from B(0,0,0) to A(2,1,3) if E = -4xya, – (4y + 2x2)a, + 8zaz. Use…

A: The line integral of vector field E→ along curve C is ∫C   E→.dr=∫abE→(r→(t).r→'(t)dt  where r(t)→…

Q: Evaluate the line integral, where C is the given curve.  ∫(with c in subscript) (x + yz) dx + 2x…

A:

Q: 2. Set up an integral to find the exact length of the curve defined by 1 f(r) 1 In r %3D where 1<r <…

A:

Q: Evaluate the line integral ∫Cydx+xdy along the curve y=x2 from the point O(0,0) to the point A(1,1)…

A:

Q: (a) Solve the line integral yzdx - xzdy + xyzdz on a particle that moves along the line segment from…

A:

Q: second going from (-1, 1) to (-2, 0). Compute the line integral -1dy – 1dæ.

A:

Q: Show that the line integral is independent of path by finding a function f such that Vf = F. 2xe-Ydx…

A:

Q: Evaluate Sr. (0, 2- z2).d along each of the following paths: (a) the line from (1,-1) to (1, 1), 固助…

A: Integration

Q: Integrate f(x, y, z) = x2 -y + z over the line segments from (0, 0, 0) to (1, 1, 0)  and then to the…

A:

Q: ZA So ydz + zdy+ ædz (3,4, 5) C , C2 (2,0,0) y (3,4,0)

A:

Q: Find the line integral of F = 2√√zi-4xj + 3√yk, from (0,0,0) to (1,1,1) over each of the following…

A:

Q: (a) Find the line integral of f (x, y, z) = -x + y + z over the straight-line %3D segment from (1,…

A:

Q: (10) Is the line integral ſc f(2)dz always vanishing for all holomorphic functions f on ID, where C…

A:

Q: Compute the line integral of F along the oriented smooth curve C shown below. (2,0, 2) (1,4, 3)

A: F =(ysinz)i +(xsinz)j+( xycosz)k dr =(dx)i +(dy)j+(dz)k line integralx-21=y-0-4=z-2-1=t (x,y,z)…

Q: 4. Use Green's Theorem to evaluate the line integral | 2xydx+ xy*dy, where C is the triangle with…

A: The given triangle C with the vertices (0,0),(1,0) and (1,2) is as as shown below.

Q: Find the line integral of F= 2√zi-4xj + 3√√yk, from (0,0,0) to (1,1,1) over each of the following…

A:

Q: Find the line integral of f(x,y,z) = -x + y+z over the straight-line segment from (1, 2, 3) to (0,…

A:

Q: Evaluate this integral Sc v – 2x)dy – (yx – x²)dx where C is a curve consisting of C, which is a…

A:

Q: 4. Use Green's Theorem to evaluate the line integral: f ye dx + x*y² dy, where C is the from (-2,0)…

A:

Q: Compute the integral of f(x) = 0.3 – 2x² + 40x³ from a = 0 to b = 0.8 using i. Trapezoidal Rule with…

A:

Q: Find the line integrals of F = yi + 3xj + zk from (0,0,0) to (1,1,1) over each of the following…

A:

Q: Calculate line integral F-dr where F(x, y) = (7x + y, 5x + 2y) and C is a rectangle with vertices 0,…

A:

Q: Use Green's Formula to evaluate the line integral along the positively oriented closed curve C:

A:

Q: b) Calculate the line integral of F along the boundary of the rectangle [0, 1 x[-1, 2], traversed…

A: According to the given information, consider the vector field. It is required to calculate the line…

Q: Evaluate the lino integral y+ + (4+2-x>y +2が Where C consitts of line segments from 0,-,2) to…

A:

Q: 2. We have seen that the FTC for line integrals implies fc Vf - dĩ = 0 for any smooth closed path C…

A: FTC for line integrals implies ∫C∇f·dr→=0 for any smooth closed path C and any smooth function f (a)…

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: