1. Evaluate ſſ, y dS where S is the portion of the cylinder 4x² + 2y² = 16 %3D that lies between z = 0 and z = 7.

Q: 4. Evaluate: S. yx?dx + xzydy + xz²dz, where C:x = sint, y = cost, z = t2 , (0 st< 1).

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Q: 8. Integrate by parts: a) St csc? t dt b) S x2 In x dx

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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: 10. (1 – y³) dx + (x³ + e*) dy, Č is the boundary of the region between the circles x² + y² = 4 and…

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Q: The differ en tial equation yʻz + 2y'e* = 0 is (a) linear (b) separable (c) homogeneous (d) exact

A: The given differential equation is: y4x+2y'ex=0

Q: Evaluate I = , dx dy where R is the region that lies inside the circle x2 + (y – 2)2= 4 and outside…

A: We have to find ∬Rdxdy where R is the region that lies inside the circle x2+y-22=4 and outside the…

Q: Which of (a)–(c) is not equal to s(x, y, z) dz dy dx? IIT ra, y. 2) dy dx dz (a) 13 III ra,y. z) dz…

A: Explanation: Given that,  ∫01∫34∫67f(x,y,z)dzdydx   Concept Used: In ∫ba∫cd∫efg(x,y,z)dxdydzlimit…

Q: Find the most general antiderivative. 4t dt O A. 12t° + 6. O B. +t+C O C. 8t++C OD. +C 18 2/9

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Q: Given a level suface 4x2y2 + 5xy = 4 with y = y(x). Find! dy at the point (3, 2). Give your answer…

A: A detailed solution is given below

Q: Calculate the following antiderivatives: (8t – 5t° – 1) dt = |+C. 1 (b) + 6.5Vu) du =+0 +C. u!/4

A: Given that: (a)∫(8t-5t8-1)dt(b)∫1u1/4+6.5udu(c)∫12x5dx

Q: Given z = 17 In(t) and y = 289 t then fir dy d'y dx?

A: Differentiate given function

Q: Determine the solution of the following homogeneous DE: sec (G) -y ]dx +xdy

A: Given differential equation is x secyx - y dx + x dy = 0

Q: D eter mine Paricular volution ( 2xy)dy = (y²-x²)dy j x=I,y= -2

A: In this question we have to determine the particular solution.

Q: 3- show that the diff. equ. is homogeneous and Solve it (2x +34) dx + (y-x)dy = 0

A: To find the solution of the given differential equation(DE).

Q: Ⓒ Given SS² +8 dy dx. Reverse the order of in tegration and evaluate. Draw the region.

A: In the given problem first we draw the region then reverse the order of integration and after that…

Q: Given z= 2 cost+6 and y= 5 sin t dz (i) dt dy (ii) dt dy (iii) da

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Q: y' = xy" %3D

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Q: Calculate . (x² + x²)e²as, where S is the cylinder with equation x² + y² = 9 for 0 < z < 10.

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Q: The differ ential equation 2 y e =0 is b. Linear homogeneOUS d. separable

A: The differential equation is given by y4x+2y'ex = 0 To evaluate : The type of differential equation.…

Q: given: W = x² + 2y² + 2² x = cos (st) : y - 2s int 35² √+ chain role, find 3 × 502 8 using dw Ət

A: To solve the following problem

Q: What is the integrating factor of the linear DE of order one ydx + (3x – 2y+ 5) dy=0? 1 A x² B y3 C…

A: Given linear differential equation is y dx+(3x-2y+5)dy=0 ⇒dxdy+3xy-2+5y=0⇒dxdy+3xy=2-5y Which is…

Q: зхе-МХxcos(Nx) 5- a. Differentiate: y = M csc(Nx)+Nx

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Q: Find Homogeneous DE y'-ay=0

A: Given that: y'-ay=0

Q: Calculate the following antiderivatives: 12t 2t° – 2 dt = +C. (b) u9/4 + 5yū du = +C. dz = +C.

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Q: If y =Aex +Be™ d'y -(m+n) dy. Show that + mny = 0 dx

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Q: Find a solution to the SDE dX; = X,(ydX; + tdt), where y ER is a constant and Xo = 1.

A: The given equation is dXt=XtγdXt+tdt. The given initial condition is X0=1. The integration formula…

Q: 5) y ² ( x dy + ydx) = (xy + 2) dyz

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Q: 2 dy + y? = ty %3D dt

A: We have to find the general solution of the given differential equation. We can rewrite the…

Q: 3- show that the diff. equ. is homogeneous and Salve it (2x+34) dx +(y-x)dy = 0

A: To solve the given differential equation

Q: 1 Evaluate dV, where E lines between the spheres a2 +3? + 22 = 1 and E 22 + y2 + z2 = 9 in the first…

A: Solution

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: Find the yacral Selution Sulution 2. d 2. dy +y=D0

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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: If D = 4x'a - 3za, + 2ya_C/m², find the following: V•D a Answer. the total charge lying within the…

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Q: Given z=3 In(t) and y=9t² then find dy de dy da %3D

A: We use parametric differentiation

Q: 25. Consider the equationdy/ dx = (y − 4 x)/( x − y) .a.Show that equationcan be rewritten as (30)…

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Q: Find the integrating factor for ty' + 2y = t5

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Q: A 3 cm length silver bar with a constant cross section area 1 cm (density 10 g/cm³, thermal…

A: The equation is u t…

Q: 1. Draw a sketch of the cylinder z= ==V36 – 4y2.

A: Given: z=1336-4y2

Q: [" In(y+z) e dxdzdy = [Show your solution.]

A: Given that,

Q: d(xy) = ? tot al dofferentral ? Zryckt x dy=9 )totol pferibral?

A: Given

Q: Find the antiderivative for each. tunction WLen C=0. a) hlx) = xt

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Q: dy + y = ť°, dt

A: Given differential equation is: dydt+y=t2 The above equation is linear in y. The integrating factor…

Q: Evalate The following Impreper integrols ロS dx I+ X2 eや。 2 え 3] X2 Jx (X-リ% dx (Xーリ(メー4)

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Q: H x,y E IR Ix+y I \x\ Ix + yl</xl. + ly!

A: Given               ∀ x,y ∈ℝ , x+y≤x+y

Q: Find the particular antiderivative that satisfies the following conditions: dy 4а 2 + 6а — 2; у(1) —…

A: We will separate the variable and the integrate as follows

Q: Find the specific antiderivative that satisfies the following conditions: dy = 3x dx -2 + 6x - 7;…

A: Given Data: Differential equation: dydx=3x-2+6x-1-7 ......(1) Condition: y(1)=8   The antiderivative…

Q: . Find the family of solutions of the following equation using integrating factors found by…

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Q: given FCX)= x° t use definatron Of desivative

A: Given: fx=x2+1x Definition of derivative: f'x=limh→0fx+h-fxh

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.