Consider the following: the vector fie ld F = (cos (xy 2) – xy 2sin (xy 2))i – 2x 2y sin (xy 2)j %3| the curve C with r(t) = (e t, e t+1), – 1

Q: Evaluate the line integral, where C is the given curve. xy dx + (x − y) dy, where C consists of…

A: In the above line integral there are two segments which needto be seperated. First (0,0) to (1,0) so…

Q: Evaluate the line integral / Vxy° +3 dx +6xy" dy, where C any curve joining (1, -1) to (16, 1).

A:

Q: Evaluate the line integral I of the one-form w = xy dx + 12y dy over the parametric curve a : [0,…

A: Given: One-form ω=xydx+12ydy and the parametric curve α:0, ln2→ℝ2 defined by α(t)=e2t, e-2t. To…

Q: Consider: | Kysin x, cos(xy)>dr, C: r(t)= (t2, t), te[0,21] Which one of the following "regular"…

A:

Q: Exercise 10.6 Evaluate the line integral using a method of your choice. e¯*(cos y dx + sin y dy),…

A:

Q: Calculate the integral of f(x, y, z) = 3x² + 3y² + zª over the curve c(t) = (cos t, sin t, t) for 0…

A:

Q: 3. Evaluate the line integral f. F· dr where F = (ev+ye")i+(xe"+e")j and C is the parameterized…

A:

Q: Vy: y(0) = 1 dx dy Vy: y(0) = 0 %3D %3D dx dy - x-y; y(2) = 2 %3D dx dy VX-y; y(2) = 1 %3D dx

A:

Q: Evaluate the line integral, where C is the given curve. z2 dx + x2 dy + y2 dz, C is the line segment…

A:

Q: Use Green's Theorem to evaluate the line integral along the given positively oriented curve.  xy2…

A: Our Aim is to evaluate the line integral along the given positively oriented curve xy2 dx+3x2y dy…

Q: Compute the line integral of the scalar function f(x,y,z)=2x2+8zf(x,y,z)=2x^2+8z over the curve…

A: To find the line integral, obtain dS as follows.

Q: Evaluate the line integral F(x, y) = xyi – y²j where the curve C is repre sented by R(t) =t²i+t³j…

A: Function : Fx, y = xy i - y2 jRt = t2i + t3j from (1,1) to (4, -8)

Q: Show F(x, y) = (2xy³ + 8)i + (3æ²y² + 2e2")j is conservative by finding a potential function f for…

A:

Q: 3. Evaluate the line integral Sc F. dr where F(x, y, z) = (e², xz, (x + y)) over the curve r(t) = t²…

A:

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 line integral I of the one-form w = xy dx + 12y dy over the parametric curve a : [0,…

A: Given: ω=xydx+12ydy To find: I=∫αω  where α:[0,ln2]→ℝ2 with α(t)=(e2t,e-2t) ∴x=e2t,y=e-2t Now,…

Q: Evaluate the line integral | F•ds where F = 2xyi + (x – y²)j and C is (a) y = x from (0,0) to (1,1)…

A: For the solution follow the next steps.

Q: Evaluate the line integral f, F . dr, where C is the curve given by F(t) = 3 cos ti+ 3 sin t j, 0<t<…

A:

Q: Suppose that f(x, y) = xy and that C is the curve 7 (t) = (t², t³), where 0 <t <1. Fill in the…

A:

Q: F(x, y) = (4ry³ + 3)i+ (6x²y² + 2e²w)j is conservative by finding a potential function f for F, and…

A:

Q: Given that F = (4y² +e") ï+ (xe" +8xy) j, calculate the line integral So F • dr exactly, where C is…

A:

Q: Compute the line integral of the scalar function f(x, y) = /1+ 9ry over the curve y = x° for 0 <¤ <…

A:

Q: Consider the following line integral. xy dx + xy3 dy, C is counterclockwise around the triangle with…

A:

Q: Suppose C is the curve r(t) = (6t,t°), for 0sts2, and F = (3x,3y). Evaluate F•Tds using the…

A:

Q: Let w = e" dx + 4x dy + sin(x1³) dz be a one-form on R³. Find the line integral of w going…

A:

Q: Consider the curve C in RS parameterized by r(t) = sin(2t) î + cos(t) ĵ+t k, con 0 <t < 2n An…

A:

Q: Evaluate the line integral I of the one-form w = (y + z) dx + (x + 2) dy + 10z dz over the line…

A: Given: One-form ω=y+zdx+x+zdy+10zdz and the line segment joining 0, 0, 0 and (3, 1, 0). To find:…

Q: Show F(x, y) = (4xy³ + 6)i + (6x²y² + 2e²¹)j LF is conservative by finding a potential function f…

A:

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: Use Green's Theorem to evaluate the line integral. Orient the curve counterclockwise. y10 dx + x6…

A:

Q: 8.) Find work done by F = (2x³y* + x, 2x*y³ + y) along the path r(t) = (t cos(at) – 1, sin (t))…

A: Since we have given that F = 2 x3 y4 +x, 2 x4 y3 +y   and rt=t cosπt-1, sinπ2t    drt = cosπt -t…

Q: Evaluate the line integral x° + y) ds C Where C is given by r(t) = 3ti+t°j for 0<t<1

A: The region C is given by: r(t)=3t i + t3 j  ;       for  0≤t≤1

Q: Evaluate the line integral F. dr, where F (x, y) = (3+2xy) i+ (x² – 3y²) j and C is the curve given…

A: Given problem:-

Q: 4. Integrate f(x, y, z) = Vx² + y² over the involute curve (cos t + t sin t)i + (sin t – t cos t)j,…

A:

Q: Compute the line integral of the scalar function f(x,y,z)=2x^2+8zf(x,y,z)=2x2+8z over the curve…

A:

Q: 3. Evaluate the line integral c F. dr where F(x, y, z) = (e², xz, (x + y)) with 0 ≤ t ≤1 over the…

A: The solution is given as

Q: Compute the line integral of the scalar function f(x,y,z)=2x2 +8z over the curve…

A:

Q: Evaluate the line integral | F•ds where F = 2xyi +(x – y²)j and C is (a) y=x from (0,0) to (1,1) (b)…

A:

Q: Evaluate the line integral, where C is the given curve. z2 dx + x2 dy + y² dz, C is the line segment…

A:

Q: Use Green's theorem to evaluate the line Integral for the curve C given in the figure. [2y dx + x…

A:

Q: Evaluate the line integral f F · dr, where F(x, y, z) = –2xi + 3yj + 3zk and C is given by the vecte…

A:

Q: Compute the line integral of the scalar function ƒ(x, y, z) = 2x² + 8z over the curve c(t) = (e¹,…

A:

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

A:

Q: Evaluate the line integral ſp(xi + Vyi) · dr where D is the oriented curve shown below. (1,1) y = x2…

A: We know that r=xi+yj As, the path followed is y=x2 So, r=xi+x2j Let x=t, then  r=ti+t2jdrdt=i+2tj…

Q: Q#03(a): Use Green's theorem to evaluate the line integral f. (x – 3y) dx (4x + y) dy, where C is…

A: Green's theorem

Q: -. Let F = (1 + xe-Y)i -e-j. Show that F is conservative and, hence, use the Fundamental Theorem for…

A:

Q: (2x + y)dx + (x + 2y)dy

A: Given integral is:  where C is any curve that joins the point (1,1) with the (2,2).

Q: A) if F=2yi-zj+xk evaluate line integral Fxdr along the curve x=cost,y=sint,z=2cost from t=0 to…

A: Given x = costy = sintz = 2cost As F = 2yi-zj+xk Then F(x,y,z) = 2y , -z , x Thus F(x(t) , y(t) ,…

Q: Compute the line integral [(yze²v² + y² + 2 − 2x) dx + (xze²v² + 2xy) dy + (xye²³² +x+32²) dz, where…

A: We have to compute the line integral ∫Cyzexyz+y2+z−2xdx+xzexyz+2xydy+xyexyz+x+3z2dz, where C is the…

Q: Compute the integral 10x/1+ 4y?ds over the curve C: R(t) = (t + 5, t), 0<t< 1.

A: