Let C denote the positively oriented boundary of the square whose sides lie along the lines x =+ 2 and y =+ 2. Compute | cosh z dz .

Q: Find the integral surface of the linear PDE xp - yq = z which contains the circle x² + y? = 1, z=1.

A: From Lagrange method

Q: 7. D is bounded by the surface (r2 + y² + z²)² = a²(x² + y² – 22). %3D

A: We have to find volume where D is bounded by the surface (x² + y² + z²)² = a²(x² + y² — z²).

Q: Let C be the positively oriented square with vertices 0, 1, 1 + i, i. Evaluate using the definition…

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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: Compute the line integral (Баллов: 2) Jo ds, Cis the right half of circle x2 + y 16 (counter…

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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: 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: If F= 2y I- 3J +x²k and s the surface parabolic cylinder y= 8x in bounded by the planes OF the 2.…

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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 x'z ds, where C is the line segment from (0, 5, 2) to (4, 6, 5).

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Q: Compute the line integral ∫_C F · dr, where F (x, y) = 〈−x^3, xy〉 and C is part of circle x^2 + y^2…

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Q: Evaluate ſſ, Vy² – x² dA where boundary of R is the parallelogram bounded by the four lines

A: In this question, we evaluate the double integral by using the change of variable. The region is…

Q: Set up the line integral ∫c x2z ds, where c is the line segment from (1, 6, -1) to (- 4, 1, 5). (No…

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Q: Compute the line integral -xdy + ydx 2(x-y)2 +4xy where L is the circle (1-x)2 + y2 = 3 with a…

A: We can write the given line integral as ∫L y2x-y2+4xydx+-x2x-y2+4xydy, where L is the circle…

Q: Let D be the half-disc in the xy-plane given by x +y <9. Calculate e-(x2+v)dA

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

A: The line joining P1,0 and Q0,1 is     x-11-0=y-00-1=t(say)⇒x=t+1,y=-t⇒dx=dt,dy=-dt

Q: |Use Green's Theorem to evaluate | -x'ydx+xy°dy where C is the boundary of the C circle x + y =16…

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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: Let R be the trapezoid with vertices (1,-1), (3,3), (3,-3) and (9,9). Give a change of variables to…

A: Given that ,  R be the trapezoid with vertices (1,-1) , (3,3) ,(3,-3) and (9,9) . Let , the vertices…

Q: Use Stoke's Theorem to evaluate · F · dĩ where F = (xz, xy, 3xz) where C is the boundary of the…

A: Using the Stoke's theorem we have to evaluate  ∮F.→dr→ where F→=xz,xy,3xz=xzi+xyj+3xzk such that the…

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: 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 xy dV where E is bounded by the parabolic cylinders y = x², x = y² E and the planes z = 0…

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Q: 38. Calculate y = x, x = 0, and x = 1. || e* dA, where D is bounded by the lines y =x +1,

A: We have to find ∫∫D ex dA, where D is bounded by the lines y=x+1, y=x, x=0 and x=1

Q: Let C be the positively oriented square with vertices (0, 0), (2,0), (2, 2), (0, 2). Use Green's…

A: Let the given square be OABC which is positively oriented.

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

A: We will solve the problem.

Q: Compute where C is the counter dockwise oriented boundary of upper-half unit disc D. a)…

A: We know that from gobern's theoram ∫Qdx +ψdy ⇒∬∂ψ∂x-∂∂ydxdy Now ∮C(y2dx+3xydy) use green's theorem…

Q: A surface x(y, z) is defined implicitly xy + yz +x'z' = 3-xZ Calculate ду

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Q: Let C be the positively oriented square with vertices (0, 0), (3, 0), (3, 3), (0, 3). Use Green's…

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Q: # 4. Evaluate the line integral | (x +2x²y) ds, where C is the upper half of the unit circle x2 + y2…

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Q: 4. Evaluate the line integral / r y* ds where C is the right half of the circle r² + y² = 16.

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Q: Evaluate .n dS and .ndS if F = (x + 2y)i– 3zj +xk, , and S is the surface of 2x + y + 2z = 6 bounded…

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Q: Find the area of the surface defined by z = xy and x2 + y2 < 7.

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Q: || x dA, R is bounded by y = x, 2x + y = 6 and the y-axis. R

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Q: 4. Let f(x, y, z) = e² + zy² and let C be a straight line segment connecting the points (4,2,1) and…

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Q: (b) Let F = (-yx², xy²). Let D be the annulus 1 < a2 + y² < 4, and let C be its boundary. Evaluate

A: Value of the given integral is

Q: Let f(x, y, z) = eª + zy² and let C be a straight line segment connecting the points (4,2,1) and…

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Q: Evaluate [[[ =*dxcydz over the sphere x +y² +=² =1.

A: to find:- to evaluate the given integral.

Q: Let W = Span{u} where u -2. Let y = |1, find the distance from y to W L3]

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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: Use Green’s theorem to evaluate the line integral 2x2y dx + 6x3 dy, where C is the circle x2+y2 = 4,…

A: The answer is as follows:

Q: b. Draw the image of the W-plane into which the region bounded by the straight lines in the Z-plane…

A: The solution is given below

Q: Find the area of the surface f(x,y)= x² + 3y over the triangular domain bounded by the x-axis, the…

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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 &amp;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 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: Sketch the reglons enclosed by the glven curves. y = Vx, y= X= 16 y y 6 4 4 3 3 2 10 15 10 15 O-1…

A: Given : y = √x  y = 1/3 x  x = 16

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…