V1-(x-1)² x +y Change the Cartesian integral . dydx into an equivalent polar integral. x² + y? 2cos(@) (A [cos (0) + sin (0)] drde 2cos(0) T/2 S [cos(0) + sin(e)] drde B) 1/2 2cos(e) [cose + sine] rdrde © L'L (D [cose + sine] rdrde -2cos(e)

Q: 29-32 Use double integration to find the area of the plane re- gion enclosed by the given curves. y…

A: Blue colour = cosh x Red colour = sinh x

Q: V2x-x² Write the integral 2 dz dy dx in cylindrical coordinates. DO NOT EVALUATE THE INTEGRAL!

A: We know that, the cylindrical coordinates in terms of the cartesian coordinates is given by,…

Q: 4-x²-y² Change the integral f ** Na*-* z² dz dx dy to spherical coordinates, you do not need to…

A: Given integral in cartesian form is, I=∫02∫04-y2∫04-x2-y2z2dzdxdy

Q: 1.5 (3-y Evaluate the integral dx dy using polar coordinate 'y

A: Given

Q: . Set up a double integral using polar coordinates that will yield the total area inside r = 2 and…

A:

Q: lo 3 Change the Cartesian integral into an equivalent polar integral. Then evaluate the polar…

A: Let's change given Cartesian integral to polar integral and then evaluate.

Q: Green’s Theorem for line integrals Use either form of Green’sTheorem to evaluate the following line…

A: By Green's theorem, we have where R is the region enclosed by curve C.

Q: 1) Consider the spherical equation given by p = 4 cos 0 sin p. Determine its corresponding equation…

A: Note : As per our guidelines we are supposed to answer only one question if there are multiple…

Q: Consider the triple integral: (ʃ_2^3 ʃ_4^8 ʃ_0^10 sin(xy-z) dxdydz).  Use a change of variables to…

A: To evaluate: ∫22∫48∫010sinxy-z dx dy dz by change of variables Consider the transformation…

Q: 3. For what value of K the velocity field in plane polar coordinates given by v, = tion of…

A: 3. The given values are,vr=K36r2-1cosθ and vθ=k36r2+1sinθ

Q: Find the surface area of the solid obtained by revolving the curve y = sin3 t, x = cos3t about the…

A:

Q: 6. Determine the volume V of the 3 dimensional lampshade formed (see diagram below) by rotating the…

A:

Q: 6. Consider the limnaçon curve with polar equation r = 2 cos 0 – 1 for 0 < 0 < 2n. (а) Set up (but…

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Q: sin(7y), 8x2+7x°cos(7y)). vative servative n of the parabola y=x2-16 from (4,0) to (-4,0). Which of…

A: We have to choose the option

Q: V2/2 4-x² 4-x2 I 2Vx2 + y2 + 3 dy dx + 2Vx2 + y? + 3 dy dæ %3D VI-x² Rewrite as an iterated double…

A: We will find out the required solution.

Q: 3 1 [fe dydx a) By using the change of order evaluate the integral 0 Jx /3 3 2 b) Use the polar…

A:

Q: 1+√/1-4y² 1-√√/1-4y² _1 = ƒ/³ f²+ √² √²/ y dx dy.

A:

Q: Evaluate the iterated integral by converting to polar coordinates V2x-x² S Sx-** Jx2 + y² dy dx So,…

A: Consider the given information. I=∫02∫02x-x2x2+y2dydx Here, fx,y=x2+y2 And, y=2x-x2x2+y2=2x

Q: Example 14. Use polar coordinates to evaluate the double integral 16-z2 dy dx (9 + x2 + y²)³/2 ° I

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Q: Sc F. dr where F(x.y) = (3x² – 4v)i + (27 + cos(,³))j

A:

Q: Suppose we have an infinite line of charge such that the linear charge distri- bution is given by…

A: As per the question we have to change the variable of integration to get it in the form of modified…

Q: 7. The area of region R bounded by the curve C is given by the line integral P-y dx + x dy 2 By…

A: The area of region R is given by: 12∮C -y dx+x dy

Q: A» = dz :1م)د %3D e²dz ミ3 す

A: The given integrals are A.             I=∮Cdzz2-1 and B.         I=∮Ceizz3dz Where C is a circle…

Q: 4-x²-y² 6. Given ſ *-* 4-y² 1 dz dx dy x²+y²+z² (b) Express the given integral as a triple integral…

A:

Q: 8) Let x= 2t - sin(2t) and y=- cos(2t). Set up an integral in parametric form to determine the…

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Q: V4-x² e-z²-y² 12. Use a change to polar coordinates to evaluate the integral dy dr.

A: Answer

Q: 9-x Consider the double integralſ, dydx. By converting to polar coordinates, the limits of…

A:

Q: F. Change the Cartesian integral to polar coordinate dy dx 171-77 حول من كان إلى يولير ار -1 0

A: Solution

Q: When transforming the previous integral applying change of variable in polar coordinates, we obtain:…

A:

Q: V9-x² Consider the double integralf, S* dydx. By converting to polar coordinates, the limits of…

A: Given- ∫-33∫09-x2dydx To Find- The value of θ by converting to polar coordiante.

Q: 1 dydx into polar integral- 4-x² Change the double integral x²+y² - drd0 2 1 drde 21 - drde 2 1 2…

A: As we know that; The polar coordinates; x=rcosθ ; y=rsinθdydx=rdrdθr=x2+y2 Given:…

Q: 2-y QI) Evaluate the integral f dx dy using polar coordinate

A:

Q: 3. Find the surface area generated by revolving the curve y = 2+ cos 0, x = sin 0 ; 0 <O< 2n about…

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Q: 3 19-x I| sin(x? +y²) dydx 2 b) Use the polar coordinate to find the integral -3

A:

Q: 9-x2 Consider the double integral dydx. By converting to polar coordinates, the limits of…

A:

Q: |4-x²-y2 1 6. Given ſ SN*-> dz dx dy x²+y²+z² (a) Express the given integral as a triple integral in…

A: Find your answer below

Q: 16–x² Choose the iterated integral that is equivalent to y dy dx in polar coordinates.

A: let x=rcosθ and y=rsinθ dydx=rdrdθydydx=r2sinθdrdθ 16-x2=yx2+y2=42=r2 r=4 Therefore r varies from 0…

Q: Evaluate the iterated integral. T/2 x sin(y) dy dx

A:

Q: 1.Evaluate the line integral ∫CF→⋅dr→ using the Fundamental Theorem of Line Integrals if…

A:

Q: İki noktalı Gauss kareleme yöntemini kullanarak 2 8x? -dx integraline yaklaşık değer bulunuz. 1+16x4…

A: We have to Evaluate the integral ∫-12128x21+16x4dx by using two-point Gaussian quadrature rule. As…

Q: 29-32 Use double integration to find the area of the plane re- gion enclosed by the given curves.…

A: Solution is in step 2

Q: z'-sin3z Evaluate | dz with C:/z| using Cauchy's Integral Formula Z-

A:

Q: The area of the surface generated by rotating the curve y = 1 = cos(3x), where 0 ≤ x ≤ , about the…

A: Let's find Area of surface,when rotated about x axis.

Q: 8. Find the area of the region bounded by the parabolas y = 6x – x² and y = x2 - 2x. 9. Find the arc…

A: according to our guidelines we can answer only one question and rest can be reposted.

Q: Evaluate the iterated integral by converting to polar coordinates   integral{integral(x^2+y^2)dy…

A: Refer to the question we need to solve the double integral by converting it into polar coordinates…

Q: 28 COMPLEX INTEGRALS rate counterclockwise around C. (Show the details.) tan : , C: \z| = 1

A:

Q: -V4-x² Consider the double integral I = S cos(x² + y²) dydx. By converting I into polar form, the…

A: According to question, Given integral, I = ∫03∫-9-x2-4-x2 cosx2 + y2 dy dx

Q: Q1: 3 1 Se dydx a) By using the change of order evaluate the integral Vx/3 3 19- b) Use the polar…

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Q: n sin y Q1.A: Evaluate: L: dydx V9-x2 C: Change to polar and evaluate the double integrals fL(x +…

A:

Q: V4 - x2 8- x2 – y 2 Convert the integral dz dy dx into an integral in spherical coordinates and…

A:

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.