. Evaluate ſſ,G(r)dA where G(r, y, 2) = arctan(y/x) and S is given by z = x² + y², with z< 9, r 2 0, y 2 0.

Q: x² sin xy 14.5.8. || - dA(x, y), if D is the region bounded by æ² = "4, x² = TY, y² = ;, and D y? =…

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Q: Find the value of c &gt; 0 such that the regionbounded by the cubic y = x(x - c)2 and the x-axis on…

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Q: 2: Evaluate [G(x, y)dx, [G(x,y)dy and [G(x, y)ds on the indicated curve C. G(x, y) = x³y² + y² sin…

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Q: w= Pcxiy,Z) = In (x+y+2) ,x= cost, and z = 4VE %3D 2X = y= Sint >use ch ain rule to Rind du in terms…

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Q: Suppose that over a certain region of space the electrical potential V is given by the following…

A: Given data:     The expression for the given electrical potential function is: V(x,y,z)=3x2-2xy+xyz.…

Q: Find the external in the isoperimetric problem of the extremum of subject to f(y² +xy'- :“)dx= 2,y…

A: Consider the functional  Iyx=∫x1x2Fx,y,y'dxwith the conditions yx1=y1 , yx2=y2and Iyx=∫x1x2Gx,y,y'dx…

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: Set up an integral to compute the arc length L of the curve C: r (t) = <글t?, 4t, v2t) from the point…

A: length of arc given below..

Q: Calculate the following line integral: Se y dx – x dy Over the four curves C given by the following…

A: To Determine :-  The line integral :  ∫C y dx - x dy over the given curves .

Q: Compute the line integral Sc 2.xy dx + x² dy along the following curves. (a) C1 along the circle x2…

A: In given question that is a conservative vector field mean exact integration in which intrgration…

Q: Find dy at (x, y) = (TT, T) for sin (xty) = 2x-ay %3D dx

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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: (b) cos(x² + y²) dy dx where R is the region 9 < x² + y? < 49.

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Q: Fined Fhe boundled by Ahe Cure area 2. x+Y= 3 %3D -33-j` and ど= %3D Ans wer

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Q: Given z = 14 In(t) and y= 196 t then find dy %3D dx dy ||

A: Given a parametric equation:                                    x = f(t) , y = g(t)  the first…

Q: When the radius r of a sphere increases with time t, the change in area dA is dt Оа. 4 3 O b. dr 8n…

A: We will find the change in area with time as following.

Q: Set up an integral to compute the arc length L of the curve C: r (t) = <글t?, 4t, v26) from the point…

A: Concept of arc length.......

Q: 27. w = xy + yz+ zx, x = r cos 0, y = r sin 0, % = r0; %3D dw dw when r = 2,0 = T/2 Ər' d0

A: Given that,  w=xy+yz+zx , x=rcosθ, y=rsinθ and z=rθ. To determine the value of ∂w∂r, when, r=2 and…

Q: 4. Evaluate the line integral 6. x²y dx – xy² dy where C consists of the arc of the circle 16 = x² +…

A: Given equation of circle is  x2+y2=16x2+y2=42 radius is 4 We have to calculate line integral ∮ cx2y…

Q: 4. If u = xy + y²z³, where x = rset, y = rs²e-t and z = r²s sint, find the value of du/ds and au/ar…

A: Since the student has posted multiple questions and does not mention any specific question, so we…

Q: Evaluate cos z dz , where C is the circle |z – 1| = 3. (Integrate counterclockwise around C.)

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Q: = f (x, y) is differentiable and x=u cos v and y =u sin v. Show that z = Oz Oz 1 + ди + и

A: According to question given that z=f(x,y) is differentiable x=u cos(v), y=u sin(v)

Q: Differentiate: y = V25 – x2 - 5 arccos (등) CCOS Oy = %3D Oy = /5+2 V5-2 %3D /5-2 /5to %3D %3D Ste/P

A: Please refer the attached image for complete solution.

Q: 4. If C is the boundary of a region bounded by x2 + y? = 16 in counterclockwise manner, use Green's…

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Q: Q3/ Evaluate: 1- SS (4x + 2) for the region enclosed by the curves y = x² and y = 2x 2- Show that…

A: 1. Consider a function of two variables, z=fx,y. The double integral of function fx,y is denoted by…

Q: # 4. Evaluate the line integral | (x +2x²y) ds, where C is the upper half of the unit circle x2 + y2…

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Q: 149. sin x cos ydx + (xy + cos x sin y)dy, where C is the boundary of the region lying between the…

A: We have to find ∫c sin x cos y dx +(xy+cos x siny)dy , where C is the boundary of the region lying…

Q: Consider the following line integral. xy dx + x²y dy, Cis counterclockwise around the triangle with…

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Q: 1. Evaluate fL, 2 dv where E lies below the plane z = 2+ 3x + 4y and above the region in the…

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Q: Evaluate       C(2x − y) dx + (x + 3y) dy. C: arc on y = x5/2 from (0, 0) to (4, 32)

A: We have to evaluate the integral ∫2x-ydx+x+3ydy along the path C, where C:y=x52 from 0,0 to 4,32.

Q: Given y=u^3−u+1 and u=1−x/1+x​, find dy/dx​ at x=0

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Q: 4. Evaluate fl, cos (2) dA where D is the region bounded by D = {(x, y)/ vLT <x < V3Ln,0 <y< x² }.

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Q: l. sin (x) dA where D is bounded by y cos x, 0 <x < /2, y = 0, and x = 0.

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Q: If y = arctan (In x), find dy/dx at x = 1/e. %3D %3D Select the correct response: O e/3

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Q: 9. xy sin yz dV, where G is the rectangular box defined by the inequalities 0 <x< T, 0 < y< 1,0 < z…

A: Given information: The integral is ∬∫xysinyzdV and the region is bounded by the rectangular box…

Q: 4. Evaluate x² dV G where G is a solid that lies in the first octant bounded by x2 + y? = 3z and z =…

A: We have solid G in the first quadrant bounded by x2+y2=3z and z=3 equating the above two equations…

Q: Find the sloution of the given DE dy y-x dx y + x y(0) = 1, h = 0.1 at x = ( "

A: Introduction: The formula of the Euler method to solve the differential equation y'=f(x,y) is,…

Q: Use Green's Theorem to evaluate Se xy dx + (1 – æy³) dy where C is shown below y (1,4) (-1,2) 2 1…

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Q: Let f (x,y) = sin(xy) + x² ln(y). fyx at (0,) is equal to: %3D А. 33 В. О С. 3 D. 1

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Q: 50. w = sin x + y²z + 2z, arcsin(t – 1), y = t³, z = 3 %3D

A: Since you have posted multiple question in this question which are not interlinked so I have solved…

Q: A)lf y = sin¯¹ (3) prove Zyy + z = 0 B)Z = 1+ Ini, find [21.

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Q: d the val 5) y= 2cos 30 6) y= 4sin 20 cos 0 7) cot F 7) y= sin (0-45) 8) y = 2 + cos 0 9) y= 3sin…

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Q: The linear approximation at I = 0 to sin(3r) is A + Br where A is: and where B is:

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Q: E). f(x, y) = sen?(r2 – 0²) + In(r0 + 0), r = Vx2 + y² 0 = arctan con -

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Q: Show that the subtangent to y = x' at x = a is equal to a/3.

A: Definition used -                            Length of subtangent =ydxdy

Q: The line integral (23 (x° + y°) ds, where C is the arc of the parabola y = x2 from (0,0) to (2, 4),…

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Q: If z=x+ iy=r(cos(0) + isin(@))then |z|=Vx²+y² =rO Z=r0 arc z=00

A: Given z=r(cos θ +i sin θ)   Modulus of complex number, |z|=x2+y2

Q: the ( gmplicit valne ) Find differenciation of az at the given poin t. Sin (y)+ sin(yn) + Sin (z+X)…

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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.