29. Let C be a simple closed smooth curve in the plane 2x + 2y + z = 2, oriented as shown here. Show that 2y dx + 3z dy – x dz 2x + 2y + z = 2 - 2 `C х depends only on the area of the region enclosed by C and not on the position or shape of C.

Q: 15. If the surface S has projection R on the xy-plane, show that F.īdS = S da dy ה.F provided |ñ -…

A: Question: If the surface S has projection R on the xy-plane, show that ∬SF¯·n¯dS=∬RF¯·n¯dxdyn¯·k¯…

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

A:

Q: 3. Compute the scalar line integral 3 y – xz ds where C is the line seg- ment from (0, 1, 1) to…

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Q: Let C be a simple closed smooth curve in the plane 2x + 2y + z = 2, oriented as shown here. Show…

A: Given, The equation of the curve is 2x+2y+z=2. The closed integral is ∮C2y dx+3zdy-x dz Let…

Q: Verify: Green's theorem in the plane for f(2x-y³) dx-xydy, where C is the boundary of the region…

A: Given information ∫c2x-y3dx-xydy Here, C is the boundary of the region enclosed by the circle…

Q: 5. . Find sSSzdv where D is the 2 solid bounded by xty'tz=4 and 2= 13x²+ 3y2. (" sce Cueam Come')

A:

Q: 5z dv, where E is bounded by the cylinder y2 + z2 = 9 and the planes x = 0, y = 3x, and z = 0 in the…

A:

Q: The area bounded by the curve z 4y+7, the y-ax y =0 and y 2 is dy3D O 18 22 11 O 15

A: Area bounded by the curve can be calculated as

Q: 39.) The solid bounded by the cylinder x+y² = 9 and the planes z= 0 and z = 3 - x. 40

A:

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

A:

Q: Verify Green's theorem in the plane for f (32 - Sy*)dz + (4y – 6zy)dy, where C is the boundary of…

A:

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: 15- Evaluate the surface integral f, xy dswhere x is the part of the plaue y = 1-x-z that lies in…

A:

Q: 5. Evaluate the line integral (r+2y) dr+x² dy where C consists of line segments from (0,0) to (2, 1)…

A: In order to solve this question by the formula of line integral .

Q: 9. Use Green's Theorem to evaluate (e* +y²) dx +(e" +x²) dy where C is the boundary of the region…

A:

Q: 5. If F=2xi - xj + y*k, evaluate ffLF av over the region bounded by the surfaces x = 0, y = 0, y = 6…

A: Given: F→=2xzi∧-xj∧+y2k∧region bounded by the surface x=0,y=0,y=6 and z=x2 ,z=4

Q: Evaluate the line integral ∫c x3 z ds, where C is the line segment from (0,8,8) to (5,7,6).

A:

Q: Evaluate the integral ∫c(2x-y)dx+(x+7y)dy along the path C, defined as y=9-x2 from (0,9) to (3,0)

A: Given ∫c2x-ydx+x+7ydy  where y=9-x2  from 0,9 to 3,0 Parametrize C by xt,yt=t,9-t2 and 0≤t≤3

Q: Let F = (-z2, 2zx, 4y - x2}, and let C be a simple closed curve in the plane x + y + z = 4 that…

A: Given F={-z2,2zx,4y-x2}. Curve in the plane (s) = x+y+z=4. Area enclosed is 16.

Q: Evaluate the line integral fe y dx + x dy where C is the parameterized path x = , y = f, 3 <iS9.…

A:

Q: Compute where Cis the counter clockwise oriented boundary of upper-half unit disc D. 0.66666666667…

A: We need to find ,  ∮ y2dx + 3xydy , over  C ,  where C is the counter clockwise oriented boundary of…

Q: Evaluate the triple integral. 4x dV, where E is bounded by the paraboloid x = 5y2 + 5z2 and the…

A: We have a paraboloid, and each cross section is a circle, parallel to the plane x = 5. We use…

Q: Verify the Green's theorem for integral x²y dx + xy dy, where C is the boundary described…

A:

Q: 8. Use Green's Theorem to calculate f, x?y³dx + (xy - y?)dy where C is the boundary of the region…

A: The given integral is ∫Cx2y3dx+xy-y2dy, where C is the boundary of the region lying between the…

Q: 17 The solid bounded by the surface y =x² and the planes y+z=4 and z = 0.

A: To find the volume of the solid

Q: Consider the region bounded by the Xxy curves J 3-2x, and y=3, %3D

A: Calculating the area of the region bounded by the curves

Q: |Evaluate the line integral: f.(r² + y) dx + (4x – cos(y)) dy where C is the boundary of the square…

A: According to Greens theorem, the line integral ∮CPdx+Qdy=∬DQx-PydA. BY the extended version of…

Q: Q7. Use Green's theorem to evaluate fy² dx + 3xy dy, where c is the boundary of the semi annular…

A:

Q: Let C be the closed, piecewise smooth curve formed by traveling in straight lines between the points…

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Q: Compute 6xy da dy, where D is bounded by the curves y = 4 – x2 and y = 2x + 4.

A:

Q: (2) Consider the region D in the ry-plane which is bounded by the following 2 4 curves y =, y = ỷ =…

A:

Q: 6- Evaluate the line integral fF dr, where F = [xy, yz, xz] and C is the boundary of the hemisphere…

A: the line integral f F dr, where F = [ry, yz, xz] and C is the boundary of the hemisphere x² + y² +…

Q: Use Green's Theorem to calculate S,x²y³dx + (xy – y²)dy where C is the boundary of the region lying…

A: Greens Theorem:

Q: The area bounded by the curve y = In(z) and the lines z = 1 and =2 above the z -azis is %3D square…

A:

Q: 4. Evaluate //F FdS, where F(x, y, z) = (2x + 2xz, -4xy², y - z2) and R R surface of the solid…

A: The given problem is to evaluate the given surface integral over the given solid surface bounded by…

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

A:

Q: Q7. Use Green's theorem to evaluate y² dx + 3xy dy, where c is the boundary o the semi annular…

A:

Q: Let D be the solid region bounded above by z = -3/x² + y² and below by z = -3. Then the projection…

A: Given Region D is bounded  above by cone  Z=-x2+y2and bounded the  below by  plane z=-3

Q: Let D be the region bounded by the parabola y = x2 and the curve y = sin x, and let P represent a…

A: If the input of the function is more than one, such functions are called multivariable functions.…

Q: Evaluate the triple integral. 4x dV, where E is bounded by the paraboloid x = 7y2 + 7z² and the…

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: 6. Evaluate the integral fSSp 4y dV if D is bounded by the elliptic paraboloid z = 3.x?+y²+1 and by…

A: Application of integration

Q: E is the region behind the surface y = 4 – x² that is in front of the region in the xy-plane bounded…

A: Given integral: ∫∬E18x dV Given that E is the region behind the surface y=4-x2 that is in front of…

Q: 1. Verify Green's theorem in the plane for 6 (3x- 8y) dx + (4y – 6xy)dy, where C is the boundary of…

A:

Q: Suppose C is a positively oriented, piecewise-smooth, simple closed curve. Verify that the result of…

A: Given    ∫Cxy2dx-x2ydy Where C is arc of parabola y=x2 from -1,1 to 1,1 and the line segment 1,1 to…

Q: Compute s F=

A:

Q: Use Green's Theorem to evaluate the line integral (у — х) dx + (2х — у) dy for the given path. C:…

A:

Q: Evaluate the triple integral. 8x dV, where E is bounded by the paraboloid x = 7y2 + 7z2 and the…

A: here, E is bounded by the paraboloid  x=7y2+7z2  and the plane x=7.    so , at    x=7 ,     y2+ z2…

Q: Verify Green's theorem in the plane for (3x² -8y²)dx+(4y-6xy)dy, where C is the closed curve of the…

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Q: 10. The solid bounded by y = 4, y – x = 1, z = x², and z = = 4.

A:

Q: 2) Use a change of variables to evaluate , zdV; where Dis bounded by the paraboloid z = 16 – x - 4y?…

A: