Consider F and C below. F(x, y) = x² i + y² j C is the arc of the parabola y = 3x2 from (-2, 12) to (-1, 3) Exercise (a) Find a function f such that F = Vf.

Q: Evaluate Se (5xy + x² + y²) dx + (x² - y)dy by using Green's theorem, where C is a closed curve that…

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Q: Use Stoke's Theorem to evaluate Į F· dr where F (x, y, =) = , and C is the triangle with C vertices…

A: given Fx,y,z=z2,y2,xy ⇒Fx,y,z=z2i^+y2j^+xyk^ and a traingle C with vertices 1,0,0 ,0,1,0 ,0,0,2 to…

Q: Let I = f x*cos(x) dx. The approximation of I using the two-point Gaussian quadrature formula is:

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Q: Let C be the parabola y = x² for 0 < x < 2. Let F(x, y) = ye"i + xe*"j.Compute the line integral

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Q: find the work done by F in moving a particle once counterclockwise around the given curve. F = 2xy3i…

A: Consider F = 2xy3i + 4x2y2j We have M = 2xy3 and N = 4x2y2

Q: At what points (x, y, z) in space are the functions in continuous? h(x, y, z) = ln (z - x2 - y2 - 1)

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Q: find the work done by F in moving a particle once counterclockwise around the given curve. F = (4x -…

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Q: Given: vector-valued function R(t) whose derivative R'(t) = (4t, 6t, —12t) (c) Determine the…

A: Here is the solution of the given problem.  To solve this problem we use simple concept of…

Q: Suppose the closed curve C in R? starts at the point (-1,1), goes along the parabola y = x2 to the…

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Q: Find the arclength of the graph of the fun ction over the indicated inter val. (a) y = 2x3/2 + 3, 0…

A: Given : Functions and their interval (a) y = 2x3/2 + 3 , 0≤x≤9 (b) y = ln(cosx), [0, π3] (c) x =…

Q: Evaluate the line integral 32xy ds, where C is the portion of the parabola y = x² from (-1,1) to…

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Q: Sketch the space curve r(t) = ⟨2 sin t, 5t, 2 cos t⟩ and find its length over the given interval [0,…

A: The given equation of space curve is  r(t) = ⟨2 sin t, 5t, 2 cos t⟩.

Q: 3. If C is the unit circle in the complex plane C, and f(2)= =², show that f(z) dz = 0 using two…

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Q: 4. Find f such that F = Vf and use it to evaluate SF. dr. (a) F(x, y) = (3 + 2xy²) î+ 2x²y ĵ, C is…

A: (4) To find: f such that F→=∇f and To Evaluate: ∫CF→·dr→. For the following Functions: (a)…

Q: 35. f(x, y) = x+2y2 -x; R is the disk x2+ y? < 4.

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Q: Find a parametrization c(t) of the line y = 5 − 2x such thatc(0) = (2, 1).

A: This question comes under the category of calculus and from the parametrization of the line here we…

Q: Given that F = (2x + y2)i + (3y – 4x)j. Verify Green's theorem for F.dr around the triangle formed…

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Q: (a) Evaluate f. (sin x + 3y²)dx + (2x – e-y*)dy, where C is the boundary of the half-disk x² + y? <…

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Q: Find the work done by F in moving a particle once counterclockwise around the given curve. F= (4x-…

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Q: = Let Co be the curve formed by the intersection of the surfaces x₁ = 2x² and X3 3x1x2, and let C be…

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Q: The function f(x, y) = ¿e=y – In(y²)

A: It is partial derivative

Q: Exercise 1.3.2 Give a paramatrization for the curve curve y = x³ + ()x-¹, then deduce the length of…

A: Given curve is  y=16x3+12x-1

Q: 2. Find the work done by F in moving a particle once counterclockwise around the given curve C F =…

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Q: 1. Sketch each curve in the z-plane, and sketch its image under w = 2². (a) |z1|=1 (b) x = 1 (e) y²…

A: We are authorized to answer three subparts at a time since you have not mentioned which part you are…

Q: Find T, N, and k for the space curve r(t) = - ti - (a cosh (t/a))j, a > 0.

A: Unit Tangent. Unit Normal. Curvature.

Q: Verify Stoke's Theorem for F =( x²)i +(xz)j + (xyz)k and C is the curve of intersection between z=1…

A: Given: F =( x²)i +(xz)j + (xyz)k and C is the curve of intersection between z=1 and z² = x² + y².

Q: Evaluate F. dr where F = (x – 3y)i + (y – 2r)j and C is the closed curve in the xy plane, x = 2 cos…

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Q: F(x, y) = 2xyi + (y² – x²)j (x2 + y2)2 and C is any positively oriented simple closed curve that…

A: Given  F(x,y) = 2xyx2+y22i+y2-x2x2+y22j To Evaluate  ∫CF(x,y)dr

Q: Let F (tanx + y)ĩ + (x + secy)j and C be the boundaries of the ellipse 9y=9 in the counterclockwise…

A: Given - F→ = tanx + yi^ + x + secyj^ and C be the boundaries of the ellipse x2 + 9y2 = 9 in the…

Q: Q4: Calculate the integral . F.dr , where F 2xy i+ x²j, if the curve c, lying in the plane XY, is:…

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Q: Let f(x, y) = x³ + 2ry – y² be a func the tangent plane to f(x, y) at the poi

A: Here we have f(x,y)=x3+2xy-y2  we have to find the equation of the tangent plane to f(x,y) at point…

Q: Given F=. By using Gauss's theorem, find f F ds for a paraboloid x = 4-y² - 2² with x ≥ 0 as shown…

A: We are given  F=-3x+1+z3,y+1+x3,6z+xy So, the divergence of F is div…

Q: 5. Verify Green's Theorem in the plane for F = (xy + y²)î + (x – y)f and the curve C is as shown in…

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Q: 7. Evaluate f F dr, where F(x, y, z) = xyi+yzj+zxk and C is the arc of the twisted cubic given by x…

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Q: Find a parametrization for the curve that matches the lower side of the parabola x-1=y2. A. x =…

A: Solution:

Q: Use Green's Theorem to evaluate / F · dĩ, where F = (VE + 3y, 2x + vũ) C consists of the arc of the…

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Q: 4 - Determine the domain of the function f(1, y) = In(z² + 2y + 1) + arcsin v1- I a)D= {(1, y)|…

A: This question is related to relation and function, we will find the domain of the given function.

Q: Express x = In(3t), y = 5 – t in the form y = f(x) by eliminating the parameter. %3D y =

A: Given that x=ln(3t) and y= 5-t we need to write y=f(x) by eliminating the parameter  Now , x= ln(3t)…

Q: Let r(t)=⟨sin(t),cos(t),9sin(t)+3cos(2t)⟩. Find the projection of r(t)r(t) onto the xz-plane for…

A: Set y=0 as the answer is asked for xz plane.

Q: Show that the graph of ř = (t sin t, t cos t, t²) lies on the paraboloid z = x? + y?.

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Q: Show that the curve r(t) = ( t3 /4 - 2)i +(4/t-3)j+cos(t-2)k is tangent to the surface x3 + y3 + z3…

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Q: Ext find the anen of the ragion bounded on the leftby the Curve yx, on the right by the line y and…

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Q: b. What is fo, f (z) dz?

A: f(z) is holomorphic except at z0   where f(z) has pole at z0 we can write  f(z)=g(z)z-z0  where g(z)…

Q: Which of the following is the equation of the tangent plane to the surface z=x2 - x+3y² +y at the…

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Q: Verify Green's Theorem in the plane for F = (x² + y²)î + (x² – y²)f in the anti-clockwise direction…

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Q: Consider F and C below. F(x, y) = x2 i + y² j C is the arc of the parabola y = 5x2 from (1, 5) to…

A: Find the curl in step2 of part (b).

Q: Let F(x, y) that starts at (0, 0)and ends at (1, 2). (6æ? + 4y) 7 + (4x + 3) j. Calculate la F . dř,…

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Q: Consider F and C below. F(x, y) = (6 + 4xy²)i + 4x²yj, %3D C is the arc of the hyperbola y = 1/x…

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Q: (1) Let f(x, y) = xe" + x2. (a) Find the tangent plane of f at the point (2,0). (b) Use your answer…

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Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.