Given F = (2xze*²z – 4, cos z, x²e*z – 3ycos?z sin z) 1. You are to show that the line integral S. F. dR is independent of path 2. Compute all the potential functions for F.

Q: Please solve the line integral Se F · dĩ In where: F = y²i+2r(y+1)j Is along 2 unique paths from the…

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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: 1. What should be A for the path independent of (2,4) ry + A 2Ay – 1 dy y? dr+ (1,2) Moreover…

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Q: Calculate the following line integral: Jc Y dx – x dy Over the four curves C given by the following…

A: Note: According to the guidelines only one question will be answered at once please upload other…

Q: 1. Use Green's theorem in order to compute the line integral $ (3cos + 6y?) dx + (sin(5") + 16x³) dy…

A: Here the given integral is                               ∮C3cosx+6y2dx + sin5y+16x3dy

Q: If F = (x + y, x² - y) a= y = √, calculate fr F.Tds by using computing the line integral directly.)

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

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Q: This problem extends Example 5.14 and Exercise 5.19, allowing us to compute the following integrals…

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Q: Show that (2x In yz-5ye") dx-(5e* -x²y)dy +(x²z- +2=) dz, where C runs from (2,1,1) to (3,1,e), is…

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

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

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

Q: Evaluate the integral, if it exists.

A: Given: ∫-π4π4t4 tant2+costdt

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

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Q: Calculate the following line integral: Sey dx – x dy Over the four curves C given by the following…

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

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

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Q: 1) Evaluate the following line integral:S. (x² + y² +z²) ds where C is parametrized by :r(t) = ti +…

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Q: Consider the curve r(t) = (t, 3 cos t, 3 sin t) for t > 0. COS So r(t)dt Compute the integral above.

A: ∫0πr(t)dt=∫0π<t,3cos t,3sin t>dt               =∫0πtdt i+∫0π3cos tdt j+∫0π3sintdt k…

Q: Evaluate the integral / (8+ sin t)º cos t dt 16 using the substitution u = 8+ sin t. %3D (8+ sin t)"…

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Q: (a) Verify that the Fundamental Theorem for Line Integrals can be used to evaluate the integral Sc…

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Q: Evaluate the line integral (cos x + sin y) ds; C is the line segment from (0, 0) to (x, 67).

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Q: Use the Fundamental Theorem of Line Integrals to calculate d exactly. 7-7+4y²f arqund the top of the…

A: Given

Q: Show that for c =5 that PV sin r dr + PV | sin(x+c) dr. This shows us that when the improper…

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Q: (a) Suppose that a black hole has magnetic field F = (2a* + e" sin(2), y³ + xe*, 2z³ + e" cos(z) +…

A: TO FIND: Magnetic flux across the black hole

Q: Evaluate the integral (t³ In ti+ 3t³ sin (t) j+k) dt.

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Q: Consider the following line integral. xy dx + x²y dy, Cis counterclockwise around the triangle with…

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Q: Consider the following line integral | (ædx + ydy + (xz – y)dz) along a curve C given by r = t2²i+…

A: We will find out the required value.

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

A: Line integral will be path independent if vector field whose integral we need  is conservative…

Q: 2. Compute the following integrals: sin(it) dt; t - T 1 -dz; 1 22 (1,1) -

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Q: Evaluate the line integral H. dř; Ħ = e" ì + [æe" + In(z)] ĵ +2² k, and C is the path defined by 7…

A: Given H→=eyi^+xey+lnzj^+2yzk^ We know that dr→=dxi^+dyj^+dzk^…

Q: 1-) Assume that x, y € R and z = x + iy e C and answer the followings. Calculate the reel integrals…

A: Given: (a) ∫-∞∞ x2x4+1 dx (b) ∫-∞∞ x2(x2+1)2 dx We have to calculate the (a) and (b)

Q: Let 0 < c < 1. Compute the integral rc+ioo dz Jc-ioo sin nz' where the integration path is the…

A: We will use the basic knowledge of complex line integrals to answer this question properly, in…

Q: What is the value of the line integral on the field 5a² y F(x, y) = () i + 5æy? 2 2 along any path…

A: The given field is Fx,y=5xy22i→+5x2y2j→. Consider the function fx,y=54x2y2. Now,…

Q: Evaluate the line integral T, (“+y) dx + 4 ry dy where L is the path, -> r) = (3 t, 2) with 0 <t <1.…

A: We evaluate given integral

Q: 3 7z Set up the iterated double integral S S g(x,y) dy dx as dxdy such that: (iki katlı integrali

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Q: (b) Let F(x,y) = (e2*y+cos(7x²), xºy^+e>Y³} Use Green's Theorem to set up the line integral F. dr as…

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Q: 1. Use substitution to evaluate the following integrals: dt arctan(2t) dt (b) t/1– In't (а) 1+ 4t2

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Q: If the current is defined as I(t) = sin (t). Using the Trapezoidal approximation for the integral,…

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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 integral. | (sec?(t) i + t(t² + 1)5 j + t* In(t) k) dt 5 1 tan ti + 12 ,(2+1)j+(등교(1) 1…

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Q: is Theorem 1.6. The line integrals of f in D are independent of path if and only if f has an…

A: Given: The line integrals of f in D are independent of the path if and only if f has an…

Q: Show that the line integral / 2re" dx + (2y – x²e¯") dy is independent of path where C is any path -…

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Q: Evaluate the Line Integral through the Path ab in the below Figure. for the F = p cos o a, + z sin o…

A: The general form of a vector valued function in cylindrical coordinates is given by F=F1aρ+F2aϕ+F3az…

Q: Evaluate the line integral along the given path. (4x-7y)ds C:r = 7ti+6tj, 0 ≤ y ≤8 O-454/85 O-444√85…

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Q: This problem extends Example 5.14 and Exercise 5.19, allowing us to compute the following integrals…

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Q: Evaluate the line integral ſ. 3(x – y) ds along the path C:r(t) = ti + (2 - t)j, 0<t<1

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Q: . Evaluate the integral S (t* In ti+ 3t sin (t*) j+ ) dt. +1)

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Q: Evaluate the integrals in Exercises 9–12. 4 cos (x²) dx dy 2y 9. e dx dy y/2 10. 27 sin 7x² -dx dy…

A: Given: