Use a surface integral to find the area of the triangle T in R with vertices at (1, 1, 0), (2, 1, 2), and (2, 3, 3). Verify your answer by finding the lengths of the sides and using classical geometry. [HINT: Write the triangle as the graph z = g(x, y) over a triangle T" in the xy plane.]

Q: Evaluate the line integral fc (2 + x*y)s, where C is the half circle x + y? = 1 in the upper half…

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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 integral for y'dx + z?dy + x²dz, where C is the triangular closed path joining the…

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Q: Use Green's Theorem to evaluate the line integral along the positively oriented curve C that is the…

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Q: Evaluate the integral for y dx + z*dy + x²dz, where C is the triangular closed path joining the…

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Q: Compute the line integral of F over C. (a) F = (,-1, In x) and C is the line segment from (, 4, 2)…

A: “Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Use Green's Theorem to evaluate the line integral of F = (xº, 2x') around the boundary of the…

A: Given Fx,y=x9 i+2x j Px,y=x9, Qx,y=2x ∂P∂y=0, ∂Q∂x=2 Now ∫Cx9dx+2xdy=∫∫R∂Q∂x-∂P∂ydxdy=∫∫R2dydx

Q: _y²x I+ y? vertices (0, 0), (1,0), (1, 1), and (0, 1). 1. tan- y dx - dy, where C is the square with

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Q: Use Stoke's theorem to evaluate the integral xydx +2x 'dy-&r 'zdz where C is the boundary of the…

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Q: Evaluate the surface integral 2 = 1 which is in the first octant. ry dS, where S is the part of r+y+

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Q: Consider the surface S: z = 4 - x2 ,bounded by y = −5(x2 - 1) and y = 0, just as it is shown in the…

A: We have to find the area of the surface S, and we will use surface integral to do that.

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: Use green's theorem to calculate the integral Fdr, F(r, y) = (e* + x²y , e³ – ry² + sen(3) + 2)…

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Q: Prove that the integral of the bottom of a half ellipsoid, (x/a)^2 + (y/b)^2 + (z/c)^2 =1, is equal…

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Q: Use Green's Theorem to evalute this integral where C is a triangle with point coordinates (0,0),…

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

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Q: Q:4) Use Green theorem and evaluate the line integral ${Mdx + Ndy} Where M(x,y)=y², N(x,y)=x², C is…

A: See the details solution in below

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

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Q: Use Stoke's theorem to evaluate the integral xydx +x*dy– 4x'ydz where C C is the boundary of the…

A: Given: ∫Cxydx+x2dy-4x3ydzBoundry square surfaces z=1vertices are (1,0,1),(1,1,1) & (0,1,1) &…

Q: Evaluate the surface integral: x dS,  S is the part of the plane  12x + 6y + z = 12  that lies in…

A: From the given statement, the plane is 12x + 6y + z = 12 and it is lies in the first octant.

Q: 2/ MATH

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Q: Evaluate The Contour integral ising the pariametric rapresentalim for C f(2) = 2²-1 where cUve cis.…

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Q: 344. Use Stokes' theorem to evaluate line integral (zdx + xdy + ydz) , where C is a triangle with…

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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: Evaluate the line integral using Green's Theorem and check the answer by evaluating it directly. 51…

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Q: Evaluate the integral for Sy dx + z?dy + x²dz, where C is the triangular closed path joining the…

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Q: - Prove that the contour integral – z)dz = -12i where C is the triangle with points 0, 3i and -4…

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Q: 4 Calculate the directed curvilinear integral I where L is the perimeter of triangle ABC with…

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Q: Use Green's Theorem to evaluate the integral. Assume that the curve C is oriented counterclockwise.…

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Q: Use the residue theorem to calculate the contour integral |z|=5 dz. /zl=5 (z + 1)²(z + 2i)

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Q: O (x² – y²) dx + x dy where is the circle x + y² = 9.

A: Green's theorem states that ∮CM dx+N dy=∬R∂M∂y-∂M∂ydx dy Here M=x2-y2 and N=x Hence the integral…

Q: Compute the contour integral dz, where l'is the contour of the perimeter of the square with vertices…

A: We will do it  by dividing r into 4 paths.

Q: 21. (y² dx + x² dy)

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Q: ) Evaluate the double integral | 4xy dA, D where D is the triangular region with vertices (0,0),…

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Q: Evaluate the integral | xy ds, where C is the curve consisting of the portion of the unit circle of…

A: Let us assume thatC1:x2+y2=1Put x=t,y=1-t2 and z=0.Hence,dx=dtdy=-t1-t2dz=0and 0≤t≤1.Let C2:The line…

Q: Use Cauchy's Integral formula to evaluate the integral: Sa (z+1) e2z dz where C is a circle |z| = 3.…

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Q: Use Green's Theorem to evaluate the following line integral. Of dy-g dx, where (f,g) = (8x,5y) and C…

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Q: Evaluate the integral for y dx + z²dy + x*dz, where C is the triangular closed path joining the…

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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: (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: Use Green's Theorem to evaluate the integral |x?ydx + xydy where Cis the rectangle with vertices (0,…

A: To evaluate: The integral ∫Cx2ydx+xydy. Given information: The vertices of the rectangle is 0,0,…

Q: Let C be the unit circle {z : |z| = 1} with positive orientation. By any method, compute the…

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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: Use Stoke's theorem to evaluate the integral xydx+x'dy – 4x’ydz where C is the boundary of the…

A: Here the surface is the plane of the square with vertices A1,0,1, B1,1,1, C0,1,1 and D0,0,1.  To…

Q: 5) Using Green's theorem, convert the line integral .(6y? dx + 2xdy) to a double integral, where C…

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Q: Evaluate the line integral using Green's Theorem and check the answer by evaluating it directly.…

A: Given Integral is ∮C3y2 dx+3x2 dy Using Green's Theorem: ∮CM dx+N dy=∬R∂N∂x-∂M∂ydydx

Q: Evaluate the contour integrals cosh z dz 4z2 + 72 z +1 dz 223 – 322 – 2z and around the unit circle…

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Q: Compute the line integral of F over C. (a) F = (,-1, In a) and C is the line segment from (, 4, 2)…

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Evaluate the surface integral. v² as S is the part of the sphere x2 + y2 + z2 = 64 that lies inside…

A: A scalar function f(x,y,z) is a rule that is used to associate a number or quantity with a…