Let C be the positively oriented circle x2 +y2 =1. Use Green's Theorem to evaluate the line integral ∫C 3ydx+9xdy

Q: Let C be the positively oriented circle r? + y? =1. Use Green's Theorem to evaluate the line…

A: Just I have used the Green's theorem to change the line integral into double integral and then by…

Q: 6. For any closed surface S, prove that || CurlF.N dS=0.

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Q: Evaluate the line integral [6. 6ayⓇ ds, where C is the right half of the circle x² + y² = 4.

A: To evaluate :   ∫C6xy6ds, where C is the right half of the circle x2+y2=4 Prerequisite : We can…

Q: Consider the right triangle withvertices (0, 0), (0, b), and (a, 0), where a > 0 and b > 0.…

A: Given right angle triangle has vertices  (0, 0), (0, b), and (a, 0), where a > 0 and b > 0.…

Q: Let C be the positively oriented circle x? + y? = 1. Use Green's Theorem to evaluate the line…

A: Compare the integral ∫C19ydx+20xdy with ∫CMdx+Ndy to obtain M=19yN=20x Consider…

Q: Let r(t) be a positively oriented rectangle with corners (0,0), (0,1), (2,1) and (2,0) and F(r, y)…

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Q: Let YR be the circle of radius R with counter clockwise orientation. Prove that 2 – 1 - lim R00 YR…

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Q: Let C be the positively oriented circle x2 +y2 =1. Use Green's Theorem to evaluate the line integral…

A: C is the circle  x2 + y2 = 1 The integral to find  ∫c3ydx +9xdy

Q: By using polar coordinate, evaluate Sl, x² + y² dA, if R is the region in between circle x² + y² = 1…

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Q: Evaluate the line integral xdy- xydx over the closed path C where Cis the circle C

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Q: Calculate the line integral ∫Cy2ds, where C is a part of the circle x=acost, y=asint, 0≤t≤π/2.

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Q: Let M C R³ be the set obtained by rotating the circle {(x, 0, z) E R³ : (x – 2)² +2² = 1} about the…

A: Solution is given below

Q: 7. Use Green's Theorem to evaluate the integral f x²y dx - xy²dy where C: is the circle x² + y² = 4…

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Q: Evaluate ſſ, Vy² – x² dA where boundary of R is the parallelogram bounded by the four lines

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Q: 3) Prove thet the xy-plane with constant curva tue k= urve In is o circle of radious

A: We have to prove that a curve in the xy plane with constant curvature k=1r is a circle of radius r

Q: Find the area of the inner loop of the curve r= 1-2cos θ

A: For the inner loop of the curve 1-2cos θ=0⇒2cos θ=1⇒cos θ=12⇒θ=cos-112⇒θ=π3,5π3

Q: Compute the line integral -xdy + ydx 2(x-y)2 +4xy where L is the circle (1-x)2 + y2 = 3 with a…

A: We can write the given line integral as ∫L y2x-y2+4xydx+-x2x-y2+4xydy, where L is the circle…

Q: The intersection between the surface x + y + z = - 1 and the circle x2 + y2 = 1,z = 0 is

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Q: Evaluate the integral eiz dz, (22 + 1)2 where C is the circle |2 + il = oriented anticlockwise.

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Q: 4. Show that the set of points (z, w) e C with w2 = sin z is naturally a Riemann surface. %3D

A: Let, S=z,w∈ℂ2|w2=sinz be the set in equation. The multivalued analytic function fz=sinz can be…

Q: If C is the line segment which begins at (-2, 1) and ends at (2, 4), evaluate the line integral: et…

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Q: Use Green's Theorem to evaluate the integral (2x? – y)dx + 4xdy where C is the circle x? + y² = 9…

A: To evaluate the given integral using green's Theorem.

Q: Use the Divergence Theorem to evaluate (4x + 9y + z2) ds where S is the sphere x2 + y2 + z? = 1.

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: Let C be the circle |2- i 2 oriented counterclockwise. Evaluate the following integral 1. z(2² + 4)

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Q: Sketch the space curve r(t) = −ti + 4tj + 3tk and find its length over the given interval [0, 1] .

A: Given space curve is rt=-ti+4tj+3tk. Sketch the above space curve as shown below.

Q: Let C be the positively oriented circle x? + y? = 1. Use Green's Theorem to evaluate the line…

A: Given : C is a positively oriented circle x2+y2=1 We have to evaluate the line integral…

Q: Let C be the line segment from point (0, 1, 1) to point (2, 2, 3). Evaluate line integral yds. C

A: To evaluate the line integral

Q: Let F(x, y, z) (x - 2)2 + (y – 1)2 = 9, z 0 oriented in the counter-clockwise direction. (y? + z3,…

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Q: O By using Green's theorem, evaluate $ (In (2²) + æy² – y²) dæ + (ev + x²y – 3ry) dy, where C is the…

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Q: Use Green’s Theorem to evaluate the line integral Z C−→F . d−→r , where C is the circle x^2 + y^2 =…

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Q: Use Green's Theorem to evaluate the line integr y? dx + x • when C is the circle of radius 2…

A: Introduction: Let C be a positively oriented, piecewise smooth, simple, closed curve, and D be the…

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Q: Let C denote the positively oriented boundary of the square whose sides lie along the lines x =+ 2…

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Q: 3. Evaluate the integral *zp- where C is the circle |2| = 2.

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Q: Let f(x, y, z) = eª + zy² and let C be a straight line segment connecting the points (4,2,1) and…

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Q: Let S be a simple closed positively oriented contour whose graph is shown below. Using Cauchy…

A: Note "As per our company guidelines we are supposed to answer ?️only first 3 sub-parts. Kindly…

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Q: Use Green’s theorem to evaluate the line integral 2x2y dx + 6x3 dy, where C is the circle x2+y2 = 4,…

A: The answer is as follows:

Q: Let C be the positively oriented circle x² + y² = 1. Use Green's Theorem to evaluate the line…

A: Greens theorem

Q: Let F = -2yi + 2xj. Use the flux form of Green's Theorem to compute F·n ds where C is the positively…

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Q: Let YR be the arc of the circle {|z| = R} that lies between the ray Arg(2) = T/4 and Arg(z) = 37/4.…

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Q: 5. Let C denote the positively oriented circle z- 1= 3. Evaluate the integral: 3 z +2z dz 3. (z- 2)

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Q: Evaluate the line integral Levis where C is the half circle x² + y² = 1, y ≥ 0. x² + y² ds

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Q: Find a formula for the distance from the point P(x, y, z) to the xz-plane

A: The distance between two points x1,y1z1 and x2,y2z2: d=x2-x12+y2-y12+z2-z12 The distance from the…

Q: Evaluate the line integral ∫Cx2yds along the segment of the line y=x from the origin up to the point…

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Q: Show that a indeed perojective plane of gives 0rder BİBD (n+n+,n+),!

A: We need to prove that a projective plane of order n indeed gives a BIBD (n2+n+1, n+1, 1)

Q: Evaluate the double integral. (8x – 6y) dA, D is bounded by the circle with center the origin and…

A: The given data is: ∫∫D8x-6ydA To evaluate the double integral.

Q: Use Green's Theorem to Calculate the Line Integral ſy²dx + x²dy, where c is the triangle o vertices…

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Q: Evaluate the line integral: x dy – y da x² + y² where C is the circle x² + y? = a², a > 0, oriented…

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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: