Use Green's Theorem to evaluate F. dr, where F < a + 4y, 3x + ỹ > C consists of the arc of the curve y 1x x2 - from (0,0) to (1,0) and the line segment from (1,0) to (0,0).

Q: By using Green's theorem evaluate £ (Ary – y") dr + (** + 3ry) dy where C is a closed curve starting…

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Q: 4) Let F(x, y) = yi+xj. a) Show that F is conservative and find a function f such that F = = Vf. b)…

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Q: b) Show that F - dr is independent of its path, given that y F(r, y, 2) = 1+. + tan-2) i+ tanrj+ k.…

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Q: 1. Use the tangent plane to the surface f(x, y) = x² + y? mate f(1, 2). at the point P(-1,1,2) to…

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Q: Compute F• dr for the oriented curve specified. F(x, y) = (9, y), quarter circle x² + y2 = 1 with x…

A: Given:

Q: 4. Prove that the following curves are intersecting with a right angle: 5y-2x+y-xy= 0 2y+5x+x-x'y² =…

A: Find your answer below

Q: Let C be the parametric curve defined by x(t) = t²-t-6, [y(t) = 4t³ + 3t² − 6t. a. Determine the…

A: Given: xt=t2-t-6yt=4t3+3t2-6t To find: a) Horizontal tangent line b) Definite integral

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: Consider F and C below. F(x, y) = x2 i + y² j C is the arc of the parabola y = 2x2 from (-1, 2) to…

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Q: The line integral P F.d7 can be evaluated using Stoke's Theorem where F = (2x, -3, 1) and C is an…

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Q: Given: vector-valued function R(t) whose derivative R'(t) = (4t, 6t, —12t) (b) Reparametrize R using…

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Q: Compute F. dr for the oriented curve specified. F(x, y) = (9, y), quarter circle x² + y2 = 1 with x…

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Q: Let C be the curve y= 3ln(9-x^2), [-0.9,2.9], find the arc length C.

A: Given y=3ln9-x2 ,-0.9,2.9

Q: Find the arc length of y = -6x + 2 on [2,5] (use calculus). %3D L = 1 + f'(x)² dx

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Q: Use Green's Theorem to evaluate F. · dr, where F = (Va + 4y, 2x + and C consists of the arc of the…

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Q: Find the tangent plane at P = (xo, Yo, Zo) = (1,2, 8) to the graph of f(x,y) = 2x?y+ y² %3D

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Q: Consider F and C below. F(x, y) = (9 + 4xy2)i + 4x2yj, C is the arc of the hyperbola y = 1/x from…

A: Consider the given information: F(x,y)=(9+4xy2)i+4x2yj And C be the arc of the hyperbola y=1/x from…

Q: Sketch a smooth connected curve y = f(x) with f(-2) = 8, f(0) = 4, f(2) = 0, f'(2) = f'(-2) = 0, %3D…

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Q: Use Green's Theorem to evaluate F dr, where F = C consists of the arc of the curve y = læ – x from…

A: Given that       F = &lt; √x + 4y, 3x + √y &gt; C consists of the arc of the curve y = x - x2 from…

Q: 4. Let f(z) = 1 23 1 - and find [ f(z) dz, where C a circle of radius 2 about the origin with…

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Q: 1. Use Green's Theorem to evaluate |F. dr, where F(r, y) = (3y- esin, 7x + Vy + 1) and C is the…

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Q: Express the curve by an equation in x and y given x(t) = sin(t) and y(t) = 8 + cos² (t). a) O-x² +y…

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Q: Use Green's Theorem to evaluate | F. dř, where F = (Va + 4y, 2x + 9) C consists of the arc of the…

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

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Q: Let C be the curve consisting of line segments from (0, 0) to (1, 1) to (0, 1) and back to (0, 0).…

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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: Find the arc length of f(x) = from x = 0 to x = V3. 2 - 1, 13 a) О O + In(2 – v3)]I 2 b) OIV3 + In(2…

A: Length of Arc =∫x=ax=b1+dydx2dx

Q: Consider F and C below. F(x, y) = x² i + y² j C is the arc of the parabola y = 2x² from (-1, 2) to…

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Q: Use Green's Theorem to evaluate dr, where F-(VE+ 4y, 2+ C consists of the arc of the curve y 1z - z…

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Q: Let F= 3xyi + 2x² j and suppose C is the oriente curve shown in Figure 28. Evaluate F.ds both…

A: The line integral ∫CF·ds involves the integration over a specific path. In general surface,…

Q: Prove that all normal to the curve x =acid+at sin t, Y=a sin t-atcost.Are at a distance a from the…

A: We need to prove that normal to the given curve are at a distance of a from origin.

Q: Use Green's Theorem to evaluate the line integra Icon (y − x)dx + (2x − y)dy - - for the given path.…

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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: Use Green's Theorem to evaluate | F. dr, where F = (Va + 5y, 2x + Vỹ ) and C consists of the arc of…

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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: 3. Let f(x, y) = xy² + x°y². Find an equation for the tangent plane at (x, y) (-1,2). %3D

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Q: Use Green's theorem to evaluate S,(5xy + x² + y²) dx + (x² – y)dy where C is a closed curve that is…

A: The posted question has multiple questions but according to guidelines we will solve the first…

Q: Let F(x, y) = -z|. Let C be the curve parametrized by (t, cos(t), sin(t)), where 0 <t<. y Evaluate:…

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Q: Let C denote the curve of intersection between x = -5z° and y = 6z", starting from A(0,0,0) and…

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Q: Verify Green's Theorem in the plane for F = -3y³i+ (3x3 + cos y)j where C is the circle x2 + y? =…

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Q: . Find SF dř where F(x,y,z) = (3y?e*, 6ye* -, 1+) and C is the curve F(t) = (sin t, ,2-) for 0 stsn.

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Q: Let curve C be the arc of x2 + y2 = 16 traversed once counterclockwise from (0, 4) to (0, -4). Let…

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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 = 3x2 from (-2, 12) to…

A: Consider Fx,y=x2i+y2j. a. We have to find a function f such that F=∇f. Step 1: We have…

Q: Use Green's Theorem to evaluate / F · dř, where F = (Va + 6y, 3x + y9) C consists of the arc of the…

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Q: Let C be the closed curve starting at (0, 1) traveling along the parabola y = (x – 1)? up to the…

A: Using Green's Theorem we solve the above question.

Q: Let C be the part of the curve x? + y² = 1 with y 2 0, oriented counter- clockwise. Evaluate F. dr,…

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Q: Use Green's Theorem to evaluate F. dr, C where F = (Va + 6y, 4x + /ỹ C consists of the arc of the…

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Q: Use Green's theorem to evaluate S. (5xy + x² + y²) dx + (x² – y)dy where C is a closed curve that is…

A: According to the given information, it is required to use green’s theorem to evaluate the integral.

Q: Given: vector-valued function R(t) whose derivative R' (t) = (4t, 6t, -12t) (b) Reparametrize R…

A: Let's reparameterize vector value function R(t).

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.