a) Show that the area of a closed curve in the xy-plane can be expressed 1 5 p (xdy – ydx) 5 p (xỷ – yà)dt A = 2 where the integral is taken counterclockwise around the curve and x(t) and y(t) are parametric representations of a curve. b) What shape should a closed curve of given constant length be in order to enclose the greatest possible area?

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: Evaluate the unoriented line integral (22 + y° + z²) ds (2x² along the parametric curve a : [0, 1]…

A:

Q: Use Green's Theorem to evaluate the line integral . curve C: the boundary of the region bounded by…

A: Given that ∮Cex2dx+siny−7x2dy, where C is the region bounded by the parabola  y=x2 and x=y2. By…

Q: 6. (The fractional residue theorem) Show that sin(ar) L r(r² + 1) tz = 5(1 – e~“), a>0. iaz [Hint:…

A:

Q: с 2 0 Use Green's theorem to evaluate the line integral _cos fc (2tan x − 2y³) dx + (ä⁰⁹ y + 2x…

A:

Q: Find the area of the surface generated by e + e -y revolving the curve x = 2 0sys In 6 about the…

A: To find the surface area generated by revolving the curve x=ey+e-y2,0≤y≤ln6 about the x-axis. We…

Q: Use Green's Theorem to evaluate the line integral. dx + (2 + x) dy -y2/2 - C: boundary of the region…

A:

Q: Use Green’s Theorem to evaluate  ∫ sin 5y dx + 5x dy  around the boundary curve C of the region R,…

A:

Q: Use a double integral and Green's Theorem to find $C(x-y)dx+xydy,ke ( - ) dz + ry dy where C C is…

A:

Q: 10. Use Green's Theorem to evaluate the line integral (e" - y) dx + x cos(y²) dy, where C is the…

A: 1/2 + ( (1/2 )sin 1)

Q: Assuming that C is the counterclockwise oriented boundary curve of the unit square in the first…

A: It is an simple and easy question on vector calculus  Can be easily solve by use of green's theorem

Q: Evaluate the surface integral I F. dS when F(x, y, 2) = y²i+ 3x² j and S is the graph of - ža (1+æ²…

A: Here, Fx,y,z=y2i+3x2jz=121+x2-y2

Q: + y^5)dy] where C is the unit circle x^2 + y^2 = 1 traversed counterclockwise.

A: we have to compute the given integral ∮x5+y3dx-(x3+y5)dy     where C is unit circle x2+y2=1.C

Q: 2. Use a parametrization to express the area of the given surface as a double integral and then…

A:

Q: 2. Use Green's theorem in order to compute the line integral P (x – 4)³ dy – y³ dx where is the…

A: We need to compute the given integral using Green's theorem . ∮C(x-4)3dy-y3dx Where is the circle of…

Q: Jse Green's theorem to evaluate the ne integral (2tan x _cos 2y³) dx + (7⁰s y + 2x (πº where is the…

A:

Q: (b) Set up a definite integral that is equal t length of the portion of the parabola y which serves…

A: Introduction: The washer method is used to calculate the volume enclosed by two functions. In this…

Q: The region between the curve y = Vxe- and the x-axis for0 <r<∞ is rotated about the r-axis to create…

A:

Q: 1. Use a parametrization to express the area of the given surface as a double integral and then…

A:

Q: Evaluate the iterated integral by converting to polar cooordinates. ∬(e−x2 − y2 ) dy dx limits of…

A:

Q: 3 Use polar coordinates to evaluate the integral (3) f[/x² +y° dA Ris bounded by the semicircle y =…

A:

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

A:

Q: 1. Find the area of surface that is generated by revolving the portion of the curve y = V4- r2…

A:

Q: Suppose f(x, y, z) = x² + y² + z? and W is the solid cylinder with height 9 and base radius 5 that…

A:

Q: Consider the curve C consisting the the portion of the parabola y = x² from (2, 4) to (-2, 4)…

A: Given: 1. y=x2 Answer:(a) positively oriented since its upward and line joining is positive sides

Q: Use Green's theorem in order to compute the line integral (2sin z + 3y²) dx + (sin(4®) + 2x³) dy…

A:

Q: By using polar coordinates, evaluate the double integral -y²dx dy x² + y² over the region in the…

A: Answer is given using polar coordinates.

Q: Consider the line integral xy dx + xy° dy with C the triangle with vertices (0, 0). (1,0), and (1,…

A: Given, the line integral Where C is the triangle with vertices.

Q: Use Green's Theorem to evaluate the line integral ſ (y + e√x) dx + (2x + cos y²) dy, where the given…

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: 1. Use a parametrization to express the area of the given surface as a double integral and then…

A: Here we will cylindrical co ordinate system to parametrize the surface  x=r cosθ , y = r sinθ , z=z

Q: Use Green’s Theorem to evaluate the line integral   ∫_c  (x+ 2y^3)dx + (2x − 3y)dy   where C is the…

A:

Q: Use Green's Theorem to evaluate the line integral. (e-x²12-v) dx + (0-1²/2 + x) ay dy C: boundary of…

A:

Q: Use Green's theorem to evaluate the line integral along the given positively oriented curve. [(5y +…

A:

Q: 3. By using Green's Theorem in the plane, evaluate | (y - sinx) dx + cos x dy where C is the…

A:

Q: Evaluate the triple integral. SSLĘ 8x dV x = 2y2 + 2z2 and the plane x = 2. where E is bounded by…

A:

Q: Compute the integral „2020 (5y + x*0 ") dx + (19x – e -e-4yCos(y) sin y) dy over the portion of the…

A:

Q: Evaluate the triple integral. SSS 5x²e" dv I where E is bounded by the parabolic cylinder z = 1 - y²…

A:

Q: 6. Use Green's Theorem to evaluate the line integral fc 2xy dx + (x² + y²) dy, if C is the ellipse…

A: Let R be simply connected region with a piecewise  smooth boundary  'C'. If functions 'M' and 'N'…

Q: Verify the Green's theorem for the following integral O sinydx + (x – cosy)dy where curve C is the…

A: Answer

Q: 7. Use Green's Theorem to compute the line integral | ry? dx – x² dy, where C is the triangle with…

A: ∫xy2dx - x2dy triangle vertices are (0,0), (2,0), and (2,2)

Q: # 4. Use Green's Theorem to evaluate the line integral P (y*dx +2xy° dy), where C is the positively…

A: Given integral is, ∮Cy4dx+2xy3dy

Q: Use the Green's Theorem to evaluate the line integral f, FdT where F = (sin(x²)-2x²y) i+ (cos(y?) +…

A:

Q: Use Green's Theorem to evaluate the line integral cos (y) dx + x²sin (y) dy along CoS the positively…

A:

Q: Use Green's Theorem to evaluate the line integral , 6y² dx + 5x² dy, where Cis the boundary of the…

A: Green's Theorem: ∮C(Pdx+Qdy)=∫∫R∂Q∂x-∂P∂ydA

Q: Use Green's theorem to evaluate the line integral along the given positively oriented curve. 3y +…

A: We have to solve the given problem by Green's theorem.

Q: 1. Use a parametrization to express the area of the given surface as a double integral and then…

A:

Q: Use Green's Theorem to evaluate the line integral (y +eva) dx + (2x + cos y²) dy , where x2 and C is…

A:

Q: (4– x² – y²) dA Evaluate the double integral JI 2M+3N+2P d) where coefficients M ,N,P are the last…

A: Given   ∬C4-x2-y22M+3N+2P, where M=0,N=2,P=8. Evaluate the integral, here C is a positively oriented…