The integral ∫C[(3x2y+y2)dx+(x3+2xy)dy] is independent of the path. Evaluate the integral where C is the path given parametrically by R=ti+(t2+t−2)j for 0≤t≤2.

Q: af 11. Suppose + = 0, prove that the line integral | F. fe -dx + ду dr = -dy is inde- pendent of…

A: Here F = fy i+fxj

Q: Evaluate the line integral / (x+ 2- rom (0,0) to (3, 0), the path C2 fro co (0,0) by applying the…

A: Consider the provided question, Hello. Since your question has multiple sub-parts, we will solve…

Q: Calculate the line integral f -y²dx + xdy where ŕ is the path y² = 2x − x² such that x>1 and y > 0.

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Q: Find the line integral along the given path C. f(x-y)d: - y)dx, where C: x=t, y=3t+8, for 0sts6…

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Q: Verify Green's Theorem by evaluating both integrals ƏM r dx + x² dy= dA ду Ne for the given path. C:…

A:

Q: Evaluate the line integral along the given path: 1. So ry ds, where C : r(t) = 4ti + 3t²j, 0 < t <…

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Q: Find the line integral ye-xz ds Here, C is the path r(t) = [t,3t,-6t], 0sts In(8) ·

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Q: Verify Green's Theorem by evaluating both integrals y2 dx + x² dy = dA ду for the given path. C:…

A: In this question we can verify the Green's Theorem . i.e, ∫C M dx +Ndy = ∫∫R ∂N∂x-∂M∂ydA .

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

A: Explanation of the answer is as follows

Q: Evaluate the line integral (x+2y)dx + dy, where C consists of the path C from (0,0) to (3,0), the…

A:

Q: Use Green's Theorem to evaluate the integral (y - x)dx + (2x - y)dy for the given path. C: x = 2…

A: Definition used -                                According to the green theorem -…

Q: Use Green's Theorem to evaluate the line integral |(y - x)dx + (2x – y)dy for the given path. C: x =…

A: Let's find.

Q: Show that the line integral | 2.xe- dx + (2y – a²e¬") dy is independent of path and evaluate the…

A: Here we have to show that the given line integral is path independent and evaluate the line integral…

Q: 3. Calculate the line integral S F - d7, where F = (x²y³, x³y?) and C is the path given by C 7(t) =…

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Q: Evaluate the integral ∮C 4x3y dx + x4 dy for any closed path C.

A:

Q: Verify Green's Theorem by evaluating both integrals | y2 dx + x2 dy dA ду for the given path. C:…

A:

Q: 38. Evaluate the integral dy + x³ dx for any closed path C.

A:

Q: Use Green's Theorem to evaluate the integral (v-x) dx + (2x−y)dy for the path C. boundary of the…

A:

Q: Evaluate the integral (6x-y)dx + (x+6y)dy along the path C, defined as y = 16-x² from (0, 4) (4, 0).…

A: Answer: The solution is given below:

Q: What should be i for the path independent of (24) ´ry + A 21y – 1 dy (12) over evaluate the integral…

A:

Q: Verify Green's Theorem by evaluating both integrals v² dx + x² dy = / / dA ax for the given path. C:…

A:

Q: Show that the integral S. (3x²y + y)dx + (x3 + x)dy is independent of path, if it is, then evaluate…

A:

Q: What should be A for the path independent of (2,4) 2Ay – x dy. y? XY + A (1,2) over evaluate the…

A: Consider the provide integral,

Q: Verify Green's Theorem by evaluating both integrals ax dA ду for the given path. C: rectangle with…

A: From: (0, 0) to (5, 0)y=0, dy=0∫ y2 dx+ x2 dy =0From: (5,0) to (5, 9)x=5, dx=0∫09 x2 dy= ∫0925…

Q: Verify Green's Theorem by evaluating both integrals y² dk + x² dy = / dA ay ax for the given path.…

A:

Q: Verify Green's Theorem by evaluating both integrals Lr dx + x* dy = / dA ду x² dy ax for the given…

A: In this question we verify Green's Theorem                 i.e, verify integral  ∫C y2 dx +x2 dy =…

Q: The line integral 2xe" dæ + a²e" dy + (1 – 2²)e¬²² dz is not independent of path.

A: The line integral ∫CF·dr is independent of path if the curl of F is zero, that is, ∇×F=0. Here, The…

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

A:

Q: By using Green's Theorem, evaluate the integral (1+xy') dx +x²y dy, where R is the closed R path…

A: From the given problem:   I=∫R1+xy2dx+x2ydy   From Green's theorem:   ∫F1dx+F2dy=∬∂F2∂x-∂F1∂ydxdy…

Q: Verify Green's Theorem by evaluating both integrals AN fx2ax+x²&y-LI (x-an) = y2 dx + x2 dy dA ?х ду…

A: We can solve as follows

Q: Find the line integral along the given path C. f(x - y)dx, where C: x=t, y = 5t+7, for 0 st≤5 C √(x…

A: Topic:- application of integration

Q: Evaluate the line integral | H. dř; Ħ = e" î + [xe" + In(z)] ĵ + 22 k, and C is the path defined by

A: Given that  H→=eyi^+xey+lnzj^+2yzk^      (1)  and r→=t,1,1, t∈0,1                (2)      from…

Q: Find the line integral along the given path C. [(x - y)dx, where C: x=t, y = 3t+8, for 0 ≤t≤6 с √(x…

A:

Q: Verify Green's Theorem by evaluating both integrals ам | y2 dx + x2 dy dA ду for the given path. C:…

A: Solution is given below:

Q: 1. Evaluate the line integral of E = xax - y ay along the segment P₁ to P2 of the circular path as…

A:

Q: Verify Green's Theorem by evaluating both integrals dA [_v² dx + x² dy = [ ] (ON-OM) DA ду for the…

A:

Q: Verify Green's Theorem by evaluating both integrals aN aM y2 dx + x2 dA ду ax for the given path. C:…

A:

Q: 4. Find path integral J xdx+ ydy, if L is the segment: x-1,y= and re (0.1]

A: Given x=t, y=t2, t∈0,1

Q: Evaluate the line integral IF.då over the path C to Q (2,4) from P(0,) where F(xy) = 2xy2 + 3y f 131…

A: According to question, Given function, F→x, y = 2xy2 i^ +3y j^,We have to calculate line integral…

Q: Verify Green's Theorem by evaluating both integrals √( 1² dx + x² dy = √ √ (2x - 3M) JA dA əx ду for…

A:

Q: Verify Green's Theorem by evaluating both integrals ам √₁ y ² dx + x² dy = √₂√ √ (³x - dA ax ду for…

A:

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

A: This question is related to vector calculus solved in step 2.

Q: 3. Show that the line integral | (r + y)dx + xdy is independent of path and evaluate this integral…

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Q: Verify Green's Theorem by evaluating both integrals [ 1² dx + x² oy = √ √ (2x - 3) da dA dy əx JR…

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Q: Find the line integral along the given path C. - y)dx, where C: x=t, y = 3t + 8, for 0 sts2 = xp(K -

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Q: Verify Green's Theorem by evaluating both integrals ƏN [_y² dx + x² ay = [] (ax - 3+²) da d'A dy ay…

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Q: Verify Green's Theorem by evaluating both integrals aM | y? dx + x² dy = dA ax for the given path.…

A:

Q: Verify Green's Theorem by evaluating both integrals dx + x² dy = // | da ду дх for the given path.…

A:

Q: 21

A: Please see the below picture for detailed solution.

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: