Let F = 8xe" 7 + 4x² e° j]and G = 8(x – y) i + 4(x + y)j, Let C be the path consisting of lines from (0, 0)|to (8, 0)| to (8, 4)| to (0,0). Find each of the following integrals eхactly: (a) SF - di = (b) S. Ğ · dř =

Q: 5. Verify Green's Theorem for 0(xy +x )dx+(4x-1) dy where Cis shown below by (a) computing the line…

A:

Q: Find the line integral of x ds over C where c is the line segment from (1,0) to (0,1) using two…

A:

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

A: Given:

Q: Evaluate the line integral §. xy²dx + 2x²y dy over triangular path with vertices (0,0). (2,2) and…

A:

Q: 5. Calculate the double integral f, sin(2x) cos(3y) dA , R = [0, 1/2] x [0, 1]

A:

Q: Evaluate the integral (2x – y) dx + (x + 3y) dy along the path C. C: x-axis from (0, 0) to (0, -3)…

A:

Q: Find the solution in the form of Fourier integrals: U4-3uzz = 0, |u(x, t) bounded as x → ±0, sin r…

A: We are given the following second order partial differential equation : ut - 3uxx = 0  ;  -∞ < x…

Q: Use Green's Theorem to evaluate the line integral. ex cos(2y) dx – 2ex sin(2y) dy C: x2 + y2 = a²

A: Use Green's theorem

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

A: Consider the given integral

Q: Verify Green's Theorem by evaluating both integrals ƏN [ 1² dx + 2² dy - [ ] (™N - OM) a dA ax ду…

A:

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…

A:

Q: Q.4 Show that in the following problems, the line integrals are independent of path of integration…

A:

Q: Evaluate the line integral along the path C given by x = 2t, y = 4t, where 0 ≤ t ≤ 1. √(²x- (y - x)…

A:

Q: 2. Evaluate each of the following line integrals in two ways*. F (x, y) = (2x – cos y) i + (x sin y)…

A: 2. (a) Directly: The equation of the curve C1 is x-4+y5=1( intercept form), or, y=54x+5. The…

Q: Find the line integral of F(x, y, z) = x³i + yj+ zk along th the origin to the point (3, 2, 6). F.…

A: See the details solution in below

Q: 1. What should be A for the path independent of (2,4) xy + A 2Лу — х dy y? -dx + (1,2) Moreover…

A:

Q: 3) S. (3x² – 3yz + 2xz)dx + (3y² – 3xz + z²)dy + (3z² – 3xy + x² + 2yz)dz - Calculate the…

A: Let's find.

Q: 1. Evaluate the line integral of E-xa, -y a, along the segment P; to P; of the circular path as…

A:

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

A:

Q: 00 1 (b) Calculate the improper integral dx , or show that it (x² + 25)3/2 -00 diverges . Show your…

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 [F-ds where F = 2xyi +(x-y³)j and C is (a) y =x from (0,0) to [10] (1,1)…

A:

Q: Evaulate the line integral of c (x^2+y^2) where c is the line segment from (-1,-1) to (2,2)

A:

Q: Show that in the following problems, the line integrals are independent of path of integration and…

A:

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

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: Reduce to first order and solve. (1 - x2 )y" - 2xy' + 2y = 0, y1 = x. Use : fp dx = In [1/(1- x²)] ,…

A:

Q: 3. Compute the line integral of F(x, y) consisting of the line segment from (T,0) to (2, 5),…

A:

Q: Find the solution in the form of Fourier integrals: U₁-3 = 0, uz (0, t) = 0, |u(x, t) bounded as…

A: We have the following partial differential equation : ut - 3uxx = 0  , x,t > 0 With boundary…

Q: Compute the line integral of f(x)=2x over the curve y=x² from (0,0) to (1,1). |

A: The line integral of f(x)=2x over the curve y=x2 from (0,0) to (1,1)

Q: Q5 Use Green's theorem to evaluate the line integral (cos.x+2y)dx+(x – sin y)dy C where C is the…

A:

Q: Compute the line integral of f(x)=2x over the curve y=x^2 from (0,0) to (1,1)

A: Given query is to find the the line integral for the y = 2x over y = x2.

Q: Find the value of the line integral 6xy dx + (3x² + 4y2)dy, where C is an ellipse 16 + = 1 from (4,…

A:

Q: Use the direct method to evaluate the path integral for the work done by F(x, y) = (y, -x) along…

A: -4π

Q: Evaluate the line integral along the path C given by x = 2t, y = 6t, where 0 ≤ t ≤ 1. √((x-x) (y -…

A:

Q: 7. Use the Fundamental Theorem of Line Integrals to evaluate S(6xy, 3x² + 6y²) · dr where C is any…

A: Line integral theorem

Q: Use the Fundamental Theorem of Line Integrals to calculate 7. dr exactly. 7 = (x + 2)7 + (2y + 3)j…

A:

Q: 15. Evaluate the line integral J.(3xy* - 6xy ]dx + (4x°y³ - 3x?) dy, where C is any path from (1,2)…

A:

Q: Compute the path integral of F = ⟨ y , x ⟩ along the line segment starting at    ( 1 , 0 ) and…

A:

Q: 3. Consider the line segment L = [-1-i, 1 + i]. Calculate the value of the integral 1₁² z² dz.…

A:

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

A:

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

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: over the graph y = e*, 0 < x < 1. Compute the path integral of f(x, y) = 2y² %D

A:

Q: 5. (a) ( If ryz² + x²yz + ry²z = e*v², compute . (b) Let f(r,y) be some function. Set up the…

A: (a) The equation given in the question is: We need to evaluate  This represents the derivative of…

Q: e following volterra integral of the first kind by any X -X-2 = (1+x-t) M (t) dt - In (cosx) =…

A:

Q: 1. What should be A for the path independent of (2,4) xy + A 2Ay – x du -dp y? dx + (1,2) Moreover…

A:

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

A: