Complex Form of Green's Theorem 4.53. If C is a simple closed curve enclosing a region of area A, prove that A = Zdz. 2i

Q: (c) Let T be the triangular region, with vertices (0,0), (-3, 1) and (3, 1), shown as the shaded…

A: equation of  line passes through (0, 0) and (3,1) y - 0 = 1-03-0(x -0) y = x3 equation of parallel…

Q: 1. Consider the region R in Quadrant II bounded by the line y = -x, the y-axis, and the circles,…

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Q: e Green's Theorem to evaluate [(-y²)dx + (x²)dy if the curve C is a triangle with rtices at (0,0),…

A: Top:- application of integration

Q: Evaluate || ldAwhere R is the region in the first quadrant that is outside the circle r=1 and R…

A: Put x = r cos(theta)        y = r sin(theta).

Q: The given curve is rotated about the y-axis. Find the area of the resulting surface. In(x), 15 xs 5…

A: according to our guidelines we can answer only three subparts, or question and rest can be reposted.…

Q: Question 12 Sketch the region bounded by the following curves and etermine the centroid of the…

A: We have to solve given problem:

Q: Work Problem 2 ( a) Sketch the region of the double integral + edA where R is bounded by a = 3, y =…

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Q: A farmer wishes to enclose a region of land in the first quadrant of the Cartesian plane under the…

A: The required region is as shown below: Area of the region =1. ⇒∫x=sx=tf(x)dx=1 ⇒∫x=sx=t1xdx=1…

Q: rove that the surface area of a sphere of radius a is 4πa2 by revolving the curve y=√(a2−x2) where…

A: Given that

Q: 12.6- Surface Area 1. Find the area of the part of the surface z = 1+ 3x + 2y²that lies above the…

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Q: 16. Use one of the Green's Theorem area formulas, compute the area of the region enclosed by the…

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Q: 18) (uoe Disk Method) Find the voleime of the senerated by revoluing the region bounded bay the…

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Q: (0, 1) (0,0) (1,0) FIGURE 0.1. The triangle T of the problem. Consider the triangle T with vertices…

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Q: (c) Use Green's Theorem to evaluate 3xy dx + 2xy dy, where C is a counterclockwise oriented…

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Q: (a) Find the minimum and maximum xcoordinates of points on the cardioid r =1−cosθ.(b) Find the…

A: given,       the cardioid r= 1-cosθ

Q: Fit a least-squares parabola having the form  y = a + bx + cx2 to the given data. x 1.2 1.8 3.1 4.9…

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Q: Let D contain those points z = x+iy of the circle |z| < 2 that do not lie in the circle |z − 1| < 1.…

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Q: Determine a region of the x-y plane for which ditferentia the equation would have a unique Solution…

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Q: Consider the following. y = Vx - 2) (x - 8)2 y = 0 X = 2 X = 8 (a) Use a graphing utility to graph…

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Q: # 1-2: Let Rbe the region bounded by the curves y=x and y 0, x=3. beobom ai onotl a vd boeo ioinols…

A: We will solve the first part, For this we are going to use double integral to find area bounded by…

Q: s question is differential geometry. Prove that the coordinate lines of a su

A: (1).To prove that the coordinate lines of a surface patch are lines of curvature if and only if…

Q: Use Stokes's Theorem to evaluate F. dr. Use a computer algebra system to verify your results. In…

A: Here,  g(x,y)=49-x2G(x,y,z)=z-g(x,y)

Q: Using Cauchy's integral formula (chapter 14.3), integrate counterclockwise -dz z2 + 4z + 3 where C…

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Q: 1-4 Find the area of the surface generated by revolving the given curve about the x-axis.. 1. y =…

A: 1. y=7x,  0≤x≤1 Area=12×14×1         =7

Q: (P) is a circular paraboloid around z-axis and (Q) is the surface given by + += 1 This option (P) is…

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Q: 3) consider the regionnin the second quadrant betueer the circles x?+yZ=4andx2+y2=25, and above the…

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Q: 3. An R region of the XY plane is shown below: y = -8/a 2y +z = 4 y=-1/2 2y +a= -1 R If A is denoted…

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Q: 2/3 2/3 2/3 2. Find the area of the four-cusped hypocycloid x + y = a 3. Find the area bounded by y…

A: Here we have to find the area enclosed by the given curves. We will use following formulas-…

Q: 1. (1) In ry-plane, sketch the region R which is enclosed by the following: • the part of the circle…

A: The parameterization of line y=1 is x=t, y=1

Q: 1. Find the volme of sold les between planes perpendicular to the x-axis at x = -1 and x = 1. The…

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Q: Section 6.2 Volumes 1)(6pts) Find the volume of the solid obtained by rotating the region bounded by…

A: Given y=2-x^2, y=1 ; about the line y=1

Q: Question 3 A solid is formed by revolving about the y-axis, the area bounded by the curve x^3=y, the…

A: Solution :-

Q: Question 2: Let E be the solid inside the surface x² + y² + z² = 10% and above the plane z = 5. (a)…

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Q: 5) The region enclosed by the curves y = 1/x, x = 1, x = 2, and y = 0 is rotated about the x axis.…

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Q: EXAMPLE 2 Evaluate SS F. dS, where F(x, Y, z) = 7xyi + (y² + e)j + cos(xy)k and S is the surface of…

A: GivenF(x,y,z)=7xy i+(y2+exz) j+cos(xy) kand S is the surface of the region E bounded by the…

Q: 3) Consider the polar functions r, = 0 and r, = 20 a) Sketch the area between the two spirals for 0…

A: When, r = θ   ; 0≤θ≤2πx = θ*cosθy = θ*sinθ theta (rad) theta (degrees) x = theta*cos(theta)…

Q: Question 5: Find the volume of the solid bounded by y = x³ + 1 , x = 0 and x = 3 given that the…

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Q: 2X The region bounded by goc and yeek.x.ks 3. rotate e absut y-aris find the solid akitaind.

A: Note: Here, finding the volume is not mentioned but this is a typical question where finding the…

Q: 4:- Consider the image of the sphere r(0; Ø) = (a cos 0 sin Ø; a sin 0 sin Ø; a cos Ø) where 0 0…

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Q: (a) Use Green's theorem to show that if c is a loop bounding a simply connected f(0, 2) - region D,…

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Q: Evaluate the integral // (10 · x + 9 · y)dA, where R is the parallelogram with: R vertices (-3,6),…

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Q: 5-4z transformation the circle z = 1 in to a circle 4z-2 Show that the reacti on w = %3D of radius…

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Q: H-S Integrate the fallswing funCtini aver the given regionr. R: is the triangle cut from the first…

A: Evaluation of the integral

Q: The given curve is rotated about the y-axis. Find the area of the resulting surface.: y = x3/2,    5…

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Q: : The region D in the x, y plane defined by the inequalities 0 ≤ y ≤4, √√4y-1² ≤x≤ Isas √√16-₁², can…

A: We have use relation between cartesian and polar co ordinate system  x= r cosθ  , y = r sinθAlso…

Q: 5. Consider the surface obtained by rotating the graph of f(x) = 1/2, 221, around the z-axis. This…

A: We will show this using integration ∫1∞1xdx = ln x1∞ = limx→∞ln x = ∞

Q: 2. Set up (Do Not Evaluate) the triple integral /// f(a, y, 2 )dV E for the following regions. a) E…

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Q: The given curve is rotated about the y-axis. Find the area of the resulting surface. y = 5 - x2,…

A: For the given curve- y=5-x2,              0≤x≤2 Surface area is given by- A=2π∫cdfy1+f'y2dy when it…

Q: 13-28 Sketch the region enclosed by the given curves and area. 13. y = 12 – x², y= x² – 6 %3D 14. y…

A: To find the area bound by below curves