Let R be the re "gion in the +y-plone bounded by the curves g-1 the line t:4. and the イ,ロtis. Shetch R in the ty-plane. (ompute the orea of region R. Show your work

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: Given the curve y = x´ and the lines y = 0 and y=2 – x Find the moment of R with respect to the…

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Q: Evaluate the line integral [6. 6ayⓇ ds, where C is the right half of the circle x² + y² = 4.

A: To evaluate :   ∫C6xy6ds, where C is the right half of the circle x2+y2=4 Prerequisite : We can…

Q: Let Li be the straight line segment from (0.0,1) to (1,a, 8) and let F: (z+ax, ay, x> Calculate the…

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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: 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: Find the area of the part of the plane  z = ax + by + c  that projects onto a region in the…

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Q: 6 Consider points P and Q in the x-y plane, with P = (1,0) and Q (0,1). The line integral %3D %3D Q…

A: Introduction :Given, two points P and Q .P= (1,0)Q= (0,1)We have asked for the line integral of

Q: Let D be the trapezoidal region in the xy-plane bounded by the graphs of x = 0, y = 1, y = x and y =…

A: Explanation of the answer is as follows

Q: Evaluate the line integral xdy- xydx over the closed path C where Cis the circle C

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Q: Evaluate the line integral xyz ds where C is the line segment joining the points (1, – 1, 1) to (1,…

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Q: Determine a region of the xy-plane for which the DE (1+ y) y' = x²will have a unique solution whose…

A: Theorem of existence of uniqueness: If f(x,y) and dfdy are continuous on rectangular region R, then…

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

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

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Q: Evaluate the line integral / x°z ds, where C is the line segment from (0, 2,4) to

A: Line integral: - The function to be integrated is determined along a curve in the coordinate system…

Q: 2. Evaluate directly the line integral fo a dy where C is the quarter-circle from (0, 4) to (4, 0)…

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Q: Evaluate the line integral xyld x + (x²+y)dy] +. where c is the saruare farmed by the lines

A: Let us consider a line integral ∫CP dx+Qdy then it can be written as  ∫D∫∂Q∂x-∂P∂y dA  Where, P and…

Q: Determine the projection of the following surface on the xz plane. (xz trace) 4x – + = 0

A: Given equation of surface is 4x-y24+z28=0

Q: Let F(x, y) = ye"vi + xe"Vj. Evaluate the line integral F of line segments from (0,3) to (0, 0),…

A: Given function is, Fx,y=yexyi+xexyj

Q: (z+ y – 1) cos D

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Q: Let f(x, y) = 2x and C be the line segment on the plane from (0, 0) to (3, 4). Evaluate the line…

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Q: Let C be the positively oriented circle x² + y² = 1. Use Green's Theorem to evaluate the line…

A: Greens theorem states that "for closed curve C enclosed a region R then the line integral along the…

Q: Suppose D is the region in the plane bounded by the lines x + y = 0, x +y = 3, x – 2y = 0, and x –…

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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: Let D be the trapezoidal region in the ry-plane bounded by the graphs of z = 0, y = 1, y = z and y =…

A: We have to solve for the area by integration.

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

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

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

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Q: Let T be the triangular region bounded by the lines x + y = 1, x = 0, and y = 0. Evaluate the line…

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Q: For every £ Jordon region whose topological band is a segmented smooth curve oriente d counter…

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Q: Evaluate the line integral Sa y dx +z dy, where C is the line segment from (4) (-1, 1, 2) to (2,3,…

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

A: Given:

Q: • Suppose SER" is a manifold and pas. Define (to) 8:I→s Tp(S) and show that Ipcs={ock) | Curve,…

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Q: Find the area of the part of the plane defined by 3z + 2y + z 12 that lies in the first octant.

A: Given equation of plane :  3x+2y+z= 12

Q: Evaluate the line integral ∮C x dy−y dx where C is a circular, closed path defined by x^2+y^2=16

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Q: If R is a closed bounded region in the xy-plane whose boundary curve is C with arc lengths and…

A: i. Consider that F be a vector field and F→=Mi+Nj and C is a simple closed curves positively…

Q: Let C be the positively oriented circle x² + y² = 1. Use Green's Theorem to evaluate the line…

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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: Evaluate the line integral Levis where C is the half circle x² + y² = 1, y ≥ 0. x² + y² ds

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

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Q: Let R denote the region in the first quadrant of the ry-plane that is bounded by y = V4 - 1², y =…

A: Curves in the x-y plane are  y=4-x2,y=2x &y=0 Now, when you see the area enclosed under these…

Q: Evaluate the line integral y dx + x dy where C is the parameterized path x = = ² , y = t° , 3 -ydx +…

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Q: Let F(r, y) = yeyi+xe*vj. Evaluate the line integral | 1 F. dr where C is the closed path consisting…

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Q: Let R denote the region in the first quadrant of the ry-plane that is bounded by y = /4 – x², y =…

A: The definite Integral for given problem with proper reasoning has been formulated in step 2.

Q: Use Green's Theorem to Calculate the Line Integral ſy²dx + x²dy, where c is the triangle o vertices…

A: Given that, the line integral is ∫y2dx+x2dy and the vertices of the triangle is 0,0,4,0 and 4,4. By…

Q: Find the area of the part of the plane 5x + 4y + z = 20 that lies in the first octant. (Use…

A: Part of the plane equation   5x+4y+z=20⇒z=20-5x-4yPlane lies in the first octant , so x≥0 ,…