D Find the Valve of K So that the fallowing Where u= (XxXz) V= (YY) inner praduct in an.

Q: Find Oz/ou and oz/ov for z=4e* ln(y), x=ln(u cos(v)), y = usin(v) at the point (u, v)=(2.7/4).

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Q: Iffis of class c? and z= f(y.x), con u= x°y°,v = 5x+1. Verify that| :4xy kxy+ 4x³y° + 10xy- af дхду…

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Q: Consider w = x² – y², where x = s cost and y = ssin t. Find u and hu using the Cha dw at ds Rule.

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Q: Q0 Integrate 2 from x? o to L x²_2 trapezaidal rule with four

A: SolutionIntegrate x2-2 from 0 to 4using the trapezoidal rule with 4 bandsFormula of trapezoidal…

Q: dz Find when u= 0, v= 1 if z= sin(xy) + ysin(x), x = uv, y3Du² +v?. du

A: Given  z = sin(xy) +ysin(x)              x = uv and y = u2 + v2  Substituting x and y in z we have…

Q: Find the partial integral. + 3y?) dy t.

A: Given that a  integration and we solve this integration by using general formulas of integration. We…

Q: Calculate (x²+y²) dx dy area when D is the limited to x²-y²=1, x²-y²= 9, xy = 2, xy=4

A: As per the question we are given a double integral over the 2d region D bounded by the following…

Q: 6. The solid bounded by x = y², and x = 8 – 2z2 – y?.

A: To find the triple integral over the given solid bounded by x=y2,x=8-2z2-y2.

Q: 20. Show that a²F(x, y) -dx dy дх ду over the rectangle xo < x < x1, Yo < y < y1, is F(x, у) — F(x0.…

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Q: find the derivativa dy for the frurrtion y=x by using lagarithmic differatiation.

A: Given y=x5x Taking logarithmic both sides, lny=lnx5x implies lny=5xlnx

Q: d. .2 The solid bounded by the cylinder x+y = 9 and the planes z = 0 and z = 3- x. %3D

A: The solution of the problem is given in the next steps.

Q: Evaluate the integral (2x – y) dx + (x + 3y) dy along the path C. C: x-axis from (0, 0) to (0, -3)…

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Q: 28. Evaluate by Green's theorem (x - cosh y) dx + (y + sinx) dy, where C is the rectangle with…

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Q: The integral. 1 dydx is VÆ V10 – y3

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Q: To integrate y3xdx we apply integration by parts and in the form u dv, u is set as:

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Q: Evaluate S S f3zdV where E is the solid enclosed by z = 1 and z = x2 +y?

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Q: Given etye -Z+ xy - e-0, use e =0, use impicit partial denvatve to finid 2ž at the Point (1 2t

A: Explanation of the solution is given below....

Q: Evaluate f S S 4.x² + 4y² + 4z² dV, where R is the hemisphere centered at the origin, R having…

A: Given that:- ∫∫∫R 4x2+4y2+4z2 dv where R is thehemisphere centered at the origine havingradius 2.…

Q: Reverse order, then integrate SaE ) dydx. Vx \1+

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Q: (2) Find the integral (0,2,0) and (0,0,1). where D is the tetrahedron with vertices (0,0,0),…

A: Given : ∬∫x+2y+z dV             where D is  tetrahedron with vertices 0 ,0 ,0  , 1 , 0 , 0  , 0, 2 ,…

Q: Evaluate tne following double integral ! VI- 2 (2 + y)* dyd.r. -1 0

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Q: Compute for the value of constant (C) by solving the given DE using integrable combination. Let x=0,…

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Q: Compute for the value of constant (C) by solving the given DE using integrable combination. Let x=1,…

A: Given xdy-ydx=4x4dx

Q: W= Pcxiy,Z) = In (x+y+z) ,x= cost, 2. %3D 2X = Cost, y- Sint and z= 4VE >use chain rule to Rind dw…

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Q: Evaluate | CurlF nds F(x,v,z) = - yzi + xzj – xyk, and S is that part of the cylinder 4= x² + y²…

A: Given function is  Fx,y,z=yzi+xzj-xyk To evaluate ∬SCurlF·nds where S is the part of the cylinder…

Q: Find over xlapping area two equatione also gemetrical nepresentahon 2. 4'+v?=n, u?+cV- vn)=1 whore n…

A: Given curves are,…

Q: Let w=x² +3y² + 2z² where x=rcos, y=rsin 0,z=5t find dw dw ar' at

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Q: Find temperature in a rod with length L=2 and: Ut = 4uxx u(0,t) = 1,u(2, t) = 3 0 < x <1 u(x,0) =…

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Q: Compute for the value of constant (C) by solving the given DE using integrable combination. Let x=0,…

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Q: FD is the region in the first octant between the spheres x2+y2+z²=1 and x2+y2+z²=16, then dV =

A: Converting into spherical coordinates

Q: is the follecing dif Mantial opuatíon Jmeo on mot Y with x>1. ey-x

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Q: Eraluate: 1. JJR e-(n^²+yv) the eincles of padii 1& 2 dady, R is the region of between.

A: As per the our guideline, we are supposed to solve only first question. Kindly repost other question…

Q: Numerically integrate y=x-X+4 from a-0 to b= 2 using single segment trapezoidal rule. O 12 0 8 0 6 O…

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Q: The integral JJI eV 4x2 + 4y² + 4z² dV = mt, where R is the solid hemisphere with x2 + y? + z² < 11…

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Q: 1. Evaluate ſf, y dS where S is the portion of the cylinder 4x² + 2y² = 16 that lies between z = 0…

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

Q: Calculate the double integral. 8x dA, R = [0, 8] × [0, 1] 1 + xy

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Q: Assume that a body moving with velocity v encounters resistance of the form dv/dt = -kv3/2. Show…

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Q: Evaluate 7x dV where E is the region that lies inside the cylinder x2 + y² = 8 E and between the…

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Q: Determine the avarage Valve of + he Aunction fex)=Cas(z)-Sin Cx/2) PA L-2, by using The meen vaelve…

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Q: If W = xyz with x = 2t*, y = 3t-, and z = 4t¬³, what is dw dt 24 O In(t)

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Q: -H.. OProve that fcz) =z²,s coutinuous at z=z, O prove that: if Z#20 is dis continnity at Z=Zo,…

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Q: Find the line integral (4х — 5у) ds, where C is the semicircle y = 9 – x² traversed from (3,0) to…

A: Graph the semi-circle. Take x=3cost and y=3sint Now, x'=-3sint and y'=3cost The angle covered by…

Q: Find the partial integral. Y dy

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Q: Use Green's theorem to evaluate s cy? oy)dx + (y²rlo4") dy where Cis s hoon belbw, N y = 2-x² G,1)

A: The given path contains 4 curves: C1:  from (1, 1)  to  (-1, 1)C2:  from (-1, 1)  to  (-1, -2)C3:…

Q: 10 The double integral 2x dædy, where T is the triangle with vertices (0,0), (3,0) and (6,3), equals

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Q: Prove that 3x +8 Viv = dr dy: %3D where R [-1,1] x [0, 2).

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Q: dz for z = da z In(zy) + y, y = c os (x² + 1)

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Q: find the valume 9tntrat ed by y= X? on [0,31 rotating abaut thlina X-5.(shell melthod) rated

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Q: D.Prove that y y(2)= U12,9)+ jve (2,4) anelyli the loplale's as z, than Guetion u and v Satiglics

A: Consider an analytic function f(z) of the form . Then f(z) will satisfy the Cauchy-Reimann equation…

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.