. Let C be the triangle with vertices (0,0), (1,0), and (1,1), traversed counter- clockwise. Evaluate the line integral y? dx+ xdy.

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: Y – 5x |dA 100 cos x + y

A: We have to evaluate the double integral ∬100cosy-5xx+ydxdy over the region D where the region D is a…

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 2 Cx2y)dx- 2xdy along each of the following closed paths, taken…

A: Given that the integral,

Q: Question 2. Use Green's Theorem to evaluate f, 2y?dx + 5xydy, where C is the boundary of the…

A: We have to evaluate the integral ∫C2y2dx+5xydy where C is the region between the circles x2+y2=1 and…

Q: 6- Evaluate the line integral fe F dr, where F = [ry, yz, r2] and C is the boundary of the…

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Q: Evaluate I = , dx dy where R is the region that lies inside the circle x2 + (y – 2)2= 4 and outside…

A: We have to find ∬Rdxdy where R is the region that lies inside the circle x2+y-22=4 and outside the…

Q: Evaluate the line integral [(x+ y)dx+(x - y)dy, where C is the arc of the parabola y = 2x from (0,…

A: The integral in the form ∫CM(x,y)dx+N(x,y)dy involves the integration over a specific path C. In…

Q: Evaluate the line integral f e-*(cos y dx - sin y dy), where c is the rectangle with vertices (0,0),…

A: The line integral function is ∫Ce-xcosydx-sinydy. The vertices of the rectangle is 0, 0, π, 0, π,…

Q: Find the line integral ye-xz ds Here, C is the path r(t) = [t,3t,-6t], 0sts In(8) ·

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Q: QUESTION 6 Use the Green's Theorem to evaluate the line integral xfydx +xy dy. where C is the…

A: Using the given vertices we can draw the required triangle and then apply Green's theorem to…

Q: Evaluate the integral x² dx + y? dy where C consists of the arc of the circle x2 + y² = 4 from (2,0)…

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Q: the surface z = &x + 4y th=

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Q: Can you show the steps on how to do this integral?

A: Consider the provided question, (3) Evaluate the integral by using the greens theorem,

Q: 1. Let C be a path along the upper half of a circle from the point P(2,0) to the point Q(-2,0).…

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Q: State two other forms for the line integral ∫C F ⋅ n ds given that F = ⟨ƒ, g, h⟩ .

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Q: a. Which of the following surface graphs represents f(x,y) = y²? OA.

A: Given, the surface                           z=f(x, y)=y2

Q: Evaluate the line integral g. xy? dx + 2x²y dy, where C is the triangle with vertices (0,0), (2, 2),…

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Q: Let C be the boundary of a rectangle with vertices (0, 0), (-1,0),(-1,5) and (0, 5). Then ve dx +…

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Q: b. Evaluate the line integral ycos(xy)dx +(xcos(xy)+x)dy where R is the circle x'+y =9 traversed…

A: Solution Given b). ∫y cos(xy)dx+(xcos(xy)+x3) dyevaluate the line integral

Q: Find the equation of the tangent plane to *+ y? at the point (2, –3, 10) the surface z = 4 O x + 4y…

A: We have to find

Q: Consider the region enclosed by the curves y = a2 +1 and y = 2r + 1:

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Q: Evaluate the line integral, where C is the given curve. (z? +2xy)dx+ x²dy +2xzdz, where C is the…

A: The b option is true.

Q: Evaluate the integral: I = S, dz where C is the circle |z|=1. O (A) I = –27ni. O (B) I = -3Ti. O (C)…

A: By using Cauchy integral Formula-

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

A: We will find parametric equation of L and then convert whole equation in parametric form

Q: If C is the line segment from (1,3) to, (0, 0), find the value of the line integral: S(3y² i +3x}) ·…

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Q: c) Evaluate the line integral (x+2y)dx -2xdy along the following closed path, the circle x+ y= 25,…

A: In parametric form,

Q: 23 + 3 4. Evaluate the integral dz, where C is the circle (a) |z + 3| = 3, (b) |z| = 3, (c) |z| = 7.…

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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 the integral [(3x – 2y + z)ds where C is the curve, C, and C, as shown. 3 y²+ z² 2 G 2* 3…

A: Given that ∫c3x-2y+zds. The objective is to find the integral value. ∫c3x-2y+zds=∫c13x-2y+z…

Q: Let F(x, y, z) (x - 2)2 + (y – 1)2 = 9, z 0 oriented in the counter-clockwise direction. (y? + z3,…

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Q: If C is the line segment from (6,3) to, (0,0), find the value of the line integral: Se(by? i+ 3z) -…

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Q: Let M(r, y) = y – cos y and N(r, y) = x sin y. Use the Green's Theorem to calculate the line…

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Q: Evaluate the line integral 2xyz dx + x²z dy + x?y dz where C is an oriented simple curve connecting…

A: Given line integral is,

Q: Use Stoke’s Theorem to calculate the line integral ∮C(z−y)dx +(x−z)dy +(y−x)dz. The curve C is the…

A: It is given that  ∮z-ydx+x-zdy+y-xdz and vertices of the triangle are A2, 0, 0, B0, 2, 0 and C0, 0,…

Q: If C is the line segment from (1, 5) to, (0,0), find the value of the line integral: So(4y? i + 4x…

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Q: 3. Evaluate the integral *zp- where C is the circle |2| = 2.

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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: Evaluate the line integral G. dr with G(x, y) = xyi+ 2y² j, and C is the closed path comprising the…

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Q: Evaluate the line integral y² dx + x²y dy where C' is the rectangle with vertices (0,0), (5,0), (5,…

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Q: (3z+1) z'e (b) $1+2 z'e² (a) p- - dz -dz 3 (s + z7)(1– z)z?

A: Evaluate the given two integrals over the circle z=4 (a) Let fz=3z+12zz-12z+5 The singularities of…

Q: Compute the line integral f, x?dx + y²dy where the path C consists of the arc of the circle x? + y?…

A: Here we solve the given problem of integral.

Q: Let M(x, y) = y - cos y and N(x, y) = x sin y. Use the Green's Theorem to calculate the line…

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Q: If C is the line segment from (8, 1) to, (0,0), find the value of the line integral: Se(8y² i+x3) ·…

A: First we have to write the equation of the line segment C and then convert the given line integral…

Q: Calculate the double integral ∬R(x2+y2)dxdy in the circle x2+y2=2x.

A: Write the equation of circle into standard form. The radius of the circle is 1.

Q: Evaluate the line integral 2xyz dx + x²z dy + x?y dz where C is an oriented simple curve connecting…

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Q: 3. (a) Use Green's theorem to evaluate the line integral 4 y* dx – a* dy where C is the positively…

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Q: I = Examine the given integral dz ;C z-i -|= 2 the circle

A: