Find the ared of the reyion R bound ed.by and the liney=x+2 2. 2. =y-yand the line y--x ......

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: 3. Show that the line integral + 2xy) dx + x² dy + 2xz dz, is independent of path in R3, where C…

A: We will find out the required value.

Q: 2. Prove that if two straight line segments r and r'are parallel transports another straight line L,…

A: Consider the equation

Q: Evaluate the line integral S 3ydx + 6xdy where C is the straight line path from (1, 2) to (6, 5). So…

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Q: Hor what salue of e dois th. a tollowng matrex haue an Tnverse Q. 2. 2.

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Q: Suppose that l is a line. Let H be one of the two half-planes bounded by l. Give a proof that l U H…

A: Answer is mentioned below

Q: is absolutely and uniformly conyergent for all vlues of x an t in the circle 121 <B-¹

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Q: Set up the line integral ∫c x2z ds, where c is the line segment from (1, 6, -1) to (- 4, 1, 5). (No…

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Q: he area bounded by the curve y = 3Vz and the lines z = 5 and z = 9 above the r – aris is_ quare…

A: Find the area of bounded curve with line x=5 and x= 9 above x-axis

Q: 5. Conclude on the limit of f(x, y) = +v4 xy2 as (x, y) → (0,0) along paths y = mx and x = y?

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Q: 2. Let z₁ and 2₂ be two distinct points in CU{∞} that are symmetric with respect to a circle I.…

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

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Q: find the Lineartransfromation T:Manz (R)R whose matrit Mp --

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Q: Show that the closed interval [0, 1] and the unit circle S' = {(x, y)|x² + y² = 1} cannot be…

A: Suppose there exist a homeomorphism f:[0, ∞)-----&gt;R then restriction of f f|[0,∞)\{0} : [0,…

Q: Show that, in the triangular region with vertices (0,0), (1,0) and (1,1) the following double…

A: Double integral

Q: Show that 6.8) fdx + gdy = Jc.8) (12,16) • (12,16) - (28x³3 By³dx + 35ytx dy) is independent of…

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Q: Compute I ds over the path C : ř(t) = (t², 2t, t²) from (0, 0,0) to (1, 4, 4).

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Q: be chown that y,= é" and can e are Jolutions to the diffierential ニ equatiune dy dy 4. w Cy, yz)= ?…

A: Given- It can be shown that y1 = e2x and y2 = xe2x are solutions to the differential equation d2ydx2…

Q: 7 Prove : id Is perpendicular line to ra duic at its outer end point ,then t ang ent to the cirde is

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Q: graph the inequality on a coodinate plane x &gt; with line under it 2

A: To graph the given inequality x≥2.The equation x=2 has a graph that is a vertical line with all…

Q: 1)Prove that the curves o, = {|z|=2} y 0, = {|z- i|=4} are homotypic. 2)calculate O z*dz and o z*dz…

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Q: Use Green Theorem to evaluate $ xydx+e'dy where C is the path from (0,0) to (1,1) along the graph of…

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Q: Compute (ax + by² + cz) dS over the sphere x +y +z² = 1.

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Q: P(5,3) Sinh (폼+북) x +1oydy (6xdx +10ydy 5 C0,0)

A: To show: The line integral is independent of the path. Given integral:…

Q: Prove that y, + y, +2y, 56 is a valid Gomory cut for the following feasible region. X-{yeZ: 4y, +5y,…

A: We have, y2+y3+2y4≤6 X=y∈Z+4 ;X=4y1+5y2+9y3+12y4≤34X=y∈Z+4 ;X=5y2+9y3+12y4+5y2≤34X=y∈Z+4 ;…

Q: Show that the line integral 22 +2ry) dx + x? dy + 2xz dz, is independent of path in R, where C runs…

A: We will find out the required values.

Q: 3. a) Compute : ♫ (Re z − 3z) dz where ♪ is a directed line segment from z = 0 to z = i.

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

Q: For the qutegradsSEddy tringle Over the bounded by -0,4=0 and *+y=l-

A: We have to find the integral  ∬e2x+3y dx dy   over the region R bounded by curves: x=0, y=0, x+y=1

Q: graph'the Sorution to the follow imy inginequratityon animber line nur (x-2) (x+5) z O

A: Solving the problem by doing the graph.

Q: Show that f10 fdx + gdy = S6 10 5a*y²dx + 2yx³dy is independent of path •(11,–7) ,-10) • (11,–7)…

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Q: 4. Let f(x, y, z) = e² + zy² and let C be a straight line segment connecting the points (4,2,1) and…

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Q: can u help me with pullbacks and integration plz

A: Given that the integral,

Q: Show that there is no Hamiltonian path inCay({(1, 0), (0, 1)}:Z3 ⨁ Z2)from (0, 0) to (2, 0).

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Q: If C is the line segment from (1, 4, 2) to (3, 2, 1), evaluate the line integral y+ 2) ds.

A: Express the line segment in its parametric form and then use the formula below to evaluate the line…

Q: Evaluate the line integralf, (xy + z)ds along the line segment C from (1,2,3) to (-1,1,5).

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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: A chow that the wsonskian d es will neves become zex0 to8 all spal

A: Here we use the fact ex&gt; 0 ,for all real x.

Q: Find the upper bound of zdz, where Cis the line segment from 1-/ to 1+ Attach File Browse Local…

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Q: *let (Xn) a tn IR (1, 1, 1,) be sequence and d be the Usuat metric, then (X) is A-Converaent. B…

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Q: Show that, with z = Rw, for some bounded w, || z(jw)|| w llw(jw)| ||R||. = sup means that -

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Q: 3,4 show that the integsal (Xy°ty%dx + 3) Independent of the path joining the peints (1,2)and 3,4).…

A: A function F is path independent if it is conservative. And if a function F is conservative then…

Q: Find the absolite marima and minima points between [o,u] gf y : * + 2dx' sunchion is porsbl.

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Q: => For The Wbech Ualuo of E doas the Martrix イK %3D have one roal eaomalue of Mutiplicty?? k =

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Q: flz) = Im z is continuous %3D

A: The objective is to check f(z) =Im Z is continuous Or not.

Q: Use greens theorem to evaluate line interal (x2y)dx+y2dy),where c is the closed path formed be y=x…

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Q: 4-) Given a line l with direction vector and a plone jo with normal vecter Ñ show that ln=l or inP=Ø…

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Q: Prove that the integral: √ [2xyz² dx + (x²z² + z cos yz)dy + (2x²yz + y cos yz)dz ] Is independent…

A: The given integral is ∫C2xyz2dx+x2z2+zcos yzdy+2x2yz+y cos yzdz. The objective is to prove that the…

Q: Compute / (1/zy, 1/(z + y)) dr along the path from (1, 1) to (3, 1) to (3, 6) using straight line…

A: The parametric vector form of a straight line from (1,1) to (3,1) will be given by:…

Q: Suppose 4 Let that S be lines bc and u = x -y the y 571 x are related and 14 region > y = to u ar…

A: Introduction: A finite collection of equations for which we searched for the common solutions 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: