Q1: i le'dydx a) By using the change of order evaluate the integral 0 x/3 3 V9-x b) Use the polar coordinate to find the integral | sin(x+y?) dydx -3 0

Q: 2- Show that the function is analytic everywhere except at the points z +i += and on the half line…

A: The region where a complex valued function is analytic is actually its domain of definition. A…

Q: (b) Given that the integral of a function تی- قو-4/ zV² + y² + z² dzdxdy i. Sketch the projection, R…

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Q: Please solve 18 and also answer the question of what fraction of the unit circle is contained within…

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Q: 2 se Stokes's Theorem to evaluate F. dr where F(x,y,z) = −ln√√x² + y² i + arctan с -25-2x-3y over r…

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Q: с 2 0 Use Green's theorem to evaluate the line integral _cos fc (2tan x − 2y³) dx + (ä⁰⁹ y + 2x…

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Q: a) A triple integral I is written using spherical coordinates as follows, p* cos o sin o d0 dodp. i)…

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Q: The sphere of radius a centered at the origin is expressed in Cartesan coordinate as x +y² +z? = a².…

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Q: a) Let E be the solid that is above the xy-plane, enclosed by z = 18 and = 2. Evaluate the following…

A: Given integral E be a solid above xy-plane and enclosed by z=18 and z=2r2   and triple integral…

Q: (a) Given F(x, y,z) = x*y² i+ y°zj+xz'k and S is the surface of the sphere x? + y? + z? = 4. Find…

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Q: 4) Let B be a sphere with radius a and center 0 and r be the radial dis- tance. Applying Green's…

A: Consider the provided question, Let B be a sphere with radius a and center 0 and r be the radial…

Q: Assuming that C is the counterclockwise oriented boundary curve of the unit square in the first…

A: It is an simple and easy question on vector calculus  Can be easily solve by use of green's theorem

Q: a) Show that the area of a closed curve in the xy-plane can be expressed 1 5 p (xdy – ydx) 5 p (xỷ –…

A: (a) To show that  A=12∮xdy-ydx=12∮xy˙-yx˙dt Area of a closed curve in xy-plane is A=12∮xdy-ydx…

Q: a The integral that represents the length of the curve 7(t) = (2/t, cos (2t3/2), sin (2t3/2)), where…

A: length of curve

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

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Q: 1olud Evaluate dv Whese sü the 2ィz°-9,4z0 2. hemiphere

A: To evaluate : The triple integrals ∭ y2 dV , where S is the solid hemisphere, x2+y2+z2≤9 , y≥0 in…

Q: 5. Calcluate the integral below in two ways, where C is the circle centered at z = 0 of radius ,…

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

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Q: Jse Green's theorem to evaluate the ne integral (2tan x _cos 2y³) dx + (7⁰s y + 2x (πº where is the…

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Q: (b) Set up a definite integral that is equal t length of the portion of the parabola y which serves…

A: Introduction: The washer method is used to calculate the volume enclosed by two functions. In this…

Q: Consider the double integral (x +1)² + (y – 1)? = 2 R R Where R is the regron of Xy given Find the…

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Q: 12. a. Polar coordinates Show, by changing to polar coordinates, that ca sin B In (x + y²) dx dy =…

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Q: sin(z) Evaluate the integral Tz+1)(2² + 2) dz where C is the circle |z – 2i| = 1.

A: We have to evaluate the integral ∫Csinzz+1z2+2dz where C is the circle z-2i=1

Q: Q3(a) Solve the integral below by using polar coor dinate and sketch the region 2-yt 2xy dxdy J-/2-y

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Q: 3 9–? V1+x²+? dy dx dz O JO

A: Consider the integral  ∫z=03∫x=09-z2∫y=01+x2+z2dy dx dz=∫z=03∫x=09-z21+x2+z2 dxdz According to the…

Q: V-cot? u +4cot u –1 du = ? integralini hesaplayınız. sin?

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Q: 1. Let a be a positive number. Find the surface area of the solid obtained by revolving the curve…

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Q: 3z – 2 Evaluate the integral dz, where I is the z(z – 1) simple closed contour indicated in the…

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Q: 1. Two students are trying to calculate the line integral of F(x, y) = (-y, x ) over the quarter-…

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Q: 9. Which of the integrals below represents the surface area of the parametric surface: y(u, v) = u…

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Q: Evaluate the line integral Vo•dr for the following function o and oriented curve C (a) using a…

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Q: A. Using polar coordinates, evaluate the improper integral /| e-4(2²+²) dæ dy. B. Use part A to…

A: We will find out the required value.

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: 9. Which of the integrals below represents the surface area of the parametric surface: x(u, v) = u…

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

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Q: The double integral I = S Vx²+y² dydx in polar coordintes is: the double integral None of these

A: We know that : x=rcosθand,y=rsinθ Therefore, x2+y2=r2cos2θ+sin2θ=r2⇒r=x2+y2.

Q: The area of the region inside the circle r = 6 cos e and outside the cardioid r = 2(1+ cos 0), a > 0…

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Q: 7. (Cauchy's integral formula) Compute the integrals (a) [z cos z dz (2² + 1)² 2:2 - 2z +1 (b) where…

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Q: 7. (Cauchy's integral formula) Compute the integrals (») L Cos z dz dz 2:2 – 2z + 1 (b) (22 + 1)²…

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Q: 4. Let F = (3y cos(3z) + 4z)i+ (3+ sin(3z))j. Use the Fundamental Theorem of Line Integrals to find…

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Q: a2f, a2f Q1: A- Show that + əy? ду? = 0, where f(x, y) = In(x² + y²) + tan-() %3D B- Evaluate the…

A: Some Common derivatives:  1. ddxx =1 2. ddxxn =nxn-1 3.ddxsinx=cosx 4.ddxcosx=-sinx 5.…

Q: Evaluate the iterated triple integral f S Sesin(y) cos(0) dp dp de. Present the answer in exact form…

A: Evaluate the iterated triple integral ∫0π2∫02π3∫04ρ sinφ cosθ dρ dφ dθ

Q: Using the definition of the integral f, f(2)dz only, compute the integral / zdz where the closed…

A: The integral is ∫γzdz. The closed curve γ is divided into three parts: γ=C1:2,2 →(0,0)C2:(0,0)…

Q: Let F = (x,y,z) and C be parameterized by r(t) = (cos t, sin t, - 5), for 0sts2n. Evaluate cF•dr by…

A: Given that F=<x,y,z> and C be parametrized by r(t)=<cost,sint,-5> for 0<=t<=2pi .

Q: -2- R To evaluate the improper integral dy using Cauchy's Residue Theorem, we use a sector of a…

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Q: T Express the surface area of the surface obtained by rotating the graph of y = 4 sin (x) for 0 ≤ x…

A: Recall: Surface area SA=∫ab2πy1+dydx2dx We have, y=4sin(x), a=0, b=π6

Q: 3) (a) Find the intersection points of the polar curves r = 3 + 2sin0 and r = 2 (b) Determine the…

A: Given curver=3+2sinθr=2therefore2=3+2sinθ-1=2sinθ-12=sinθtherefore θ=π+π6  or θ=2π-π6 or -π6θ=7π6…

Q: 1)Evaluate the interated integral by converting to polar coordinates! A) L,L ? + y² dydx CHECK POINT…

A: 1. A) ∫-22∫-4-x24-x2x2+y2dydxy=4-x2⇒y2=4-x2⇒y2+x2=4region of this integration is circle ⇒y2+x2=4let…

Q: z2-7z+3 Using Cauchy's Integral Formula Evaluate f ,where c is the circle Iz| = 2. z+i

A: We have to evaluate the value of the integral ∫z2-7z+3z+ιdz over the circle z=2, using the Cauchy…

Q: The area of the surface generated by revolving the line x = y from x= 0 to x=2 about the y-axis:…

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