Integrate counterclockwise dz by Cauchy's formula 9. - 3 counterclockwise around the circle C : |z – 3 – 3i| = T. 2 - dz || -

Q: 12: A tangent to the circle x² + y? = 5 at the point (1,2) and th circle x? + y² – 8x + 6y + 20 - A)…

A: Given that  A tangent to the circe x^2+y^2=5 at the point (1, 2)

Q: Find the arclength of y = 2a/2 on 1 <a< 2

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Q: 2.2 Prove that Te -dz < 3 ez z2 +7+1 where C is the arc of the circle |2| = 3 from z = 3i to z = -3.

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

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Q: Find fe F dr for F = 7yi - (sin y)j on the curve counterclockwise around the unit circle C starting…

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Q: 4. Let C be the circle |z| = 2 traversed once in the positive sense cos 3z (counter clockwise).…

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Q: 1- Find all of the points where the curve r = 2 cos (-) intersects the circle of 3, radius v2…

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Q: Assume that the two points (x1 y1) and (x2.y2) lie on the graph of the straight line y = mx + b. Use…

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Q: Consider the following C counterclockwise around the circle x+y=1 from (9, 0) to (-9, 0) (a) Find a…

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Q: te the line integral c 3 == 1 from A=(2, 0) to (0, -1) Jax + 2xyay where is the arc on the ellipse

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Q: Find the arclength of y = 6x – 3 on 0 < x < 4. Give the exact answer. -

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Q: 2. Evaluateſ. a+ (z+30:) dz along C which is the circle |z + 3i| = 3, counterclockwise. %3D (z+3i)

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Q: Find the length of the curve. r(t) = cos(6t) i + sin(6t)j + 6In(cos(t)) k, 0 <t< n/4

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Q: Find Df and Duf to:- XZ on the point (,,1) inthe direction of V = 2i -3j+6 K. %3D soLiin ansi- lo

A: Directional derivative of the function.

Q: Set up and simplify (but do not evaluate) : 3x ds, where C is the arc of the circle r2 + y² = 4 from…

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Q: 4. Let C denote the circle |z – 2i| = 1, taken in the usual counterclockwise orientation. Compute by…

A: Given: Let C  denote the circle z-2i=1 z-2i=1x+iy-2i=1x+iy-2=1 So, Centre of circle will be 0,2 and…

Q: 32. (sin y, x) • dr, where C is the boundary of the triangle with vertices (0, 0), 0 ), and 2' 2

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Q: 글 36- +| de. 1 4 dz, (z+3i)³ ] Evaluate - z+3i where C is the circle |z+ 3i| = 6 + 4, 0 < 0 < 2ñ .

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Q: 2.2 Prove that e? dz] < 3 +++1 where C is the arc of the circle |2| = 3 from z = 3i to z = -3.

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Q: 3 Use Green's theorem to evaluate [x³y dx - xy²dy, where C C is the unit circle x²+y=1 with…

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Q: Find the length of the curve. r(t) (7t, 3 cos t, 3 sin t), -2 sts 2

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Q: Find the length of the curve r(t) = (cos 3t, sin 3t, 2t3/2),0 sts 3. %3D

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Q: Integrate counterclockwise around the circle |z – 6i| = 3. z(z – li)? - NOTE: Enter the exact…

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Q: Express the circle specified by x = 2 cos(0) y = 1+ 2 sin(0) in Cartesian coordinates and find the…

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Q: A- Using Simpson s Rule to estimate the area of the circle (r-4), c (1, 0) with (n=4). B- Prove…

A: The equation of a circle having center C(1,0) and radius r=4 is given by: x-12+(y-0)2=42…

Q: If C is a circle of radius 3 centered at the point ( 3, -4 ), then evaluate £(3y – ein(=) ) dz +…

A: The integral is given by ∮C3y-esinxdx+10x-siny3+ydy Where, C is a circle of radius 3 centered at the…

Q: Express the circle specified by x = 2 cos(0) y = 1+2 sin(0) in Cartesian coordinates and find the…

A: Given: x=2cos θ y=1+2sin θ Now, x2=cos θ and y-12=sin θ Squaring both: x22+y-122=cos2θ+sin2θ…

Q: 4. Let C be the circle |z| = 2 traversed once in the positive sense cos 3z (counter clockwise).…

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Q: The unit circle x² + y = 1 can be wrriten in parametric form as x = cos(0) and y = sin0. dy Find and…

A: The unit circle can be written in parametric form as x=cosθ and y=sinθ We have to find dydx and the…

Q: 4. Let C be the circle |z| = 2 traversed once in the positive sense (counter clockwise). Compute: .…

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Q: Compute ž dz +y1 z dz, +72 and where +Y1 is the segment from 1 to -1, and +y2 is the arc on the unit…

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Q: 2.2 Prove that e zp- 3 where C is the arc of the circle |2| = 3 from z = 3i to z = -3.

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Q: ez 16. Evaluate dz, where: (4z-1) cos(Tz) a) I is the circle |z| = 1/8 traversed once…

A: We need to evaluate the complex integral in the following three regions represented by Γ,

Q: Express the circle specified by x = 2 cos(0) y = 1 + 2 sin(0) in Cartesian coordinates and find the…

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Q: 2.2 Prove that e zp- 3 2+z+1 where C is the arc of the circle |=| = 3 from z = 3i to z = -3. %3!

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Q: Prove that e dz < ++1 3 where C is the arc of the circle |2| = 3 from z = 3i to z = -3.

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Q: 2. Use Green's theorem to evaluate fe e-x³ dx + (5x + sin(y²)) dy, where C is the circle x² + y² = 4…

A: Please see the below picture for detailed solution.

Q: z+i dz , where T is the circle Iz – 3i| = 1 traversed once (2) Compute counterclockwise.

A: The general form of a complex number is given by z=x+iy, where x,y are two real numbers and i=-1.…

Q: 6. Evaluate the integral . + +1-3(z-i)2) dz, where C is the circle centered at i, with (z-1)2 Radius…

A: The general form of a complex number is given by z=x+iy, where x,y are two real numbers, and i=-1.…

Q: Express the circle specified by r = 2 cos(0) y = 1+ 2 sin(0) in Cartesian coordinates and find the…

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Q: (a) Evaluate sin 2z dz, 3z -71 where C is an anticlockwise unit circle centred at the origin. 4.

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Q: Express the circle specified by x = 2 cos(0) y = 1 +2 sin(0) in Cartesian coordinates and find the…

A: Given that,   x=2 cosθ and y=1+2 sinθ

Q: (a) Evaluate sin 2z -dz, 3z -ĝi where C is an anticlockwise unit circle centred at the origin.

A: Follow the procedure given below.

Q: Find the length of the curve. r(t) = (3t, 3 cos(t), 3 sin(t)), -3 sts 3

A: For the curve r(t) = f(t) , g(t), h(t)On the interval [a, b]The length is given by L =…

Q: Find the lengthof the curve r= a sin (0/2), 0s 0S T, a>0

A: The formula for finding arc length is L = ∫0πr2+ drdθ2dθ, where r = a sin2(θ2)

Q: The unit circle x² + y=1 can be wrriten in parametric form as x = cos(0) and y = sin0. dy Find and…

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Q: Find the length of the arc of the curve x2=(y-2)3 from P(1,3) to Q(8,6).

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Q: Consider the given curve: R(f) = (2 cos f, 12f +3,-2 sin f) Reparametrize the curve using arclength…

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Q: 3 dz where C, is the top half of circle with radius (b) Compute the upper bound of 2,

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Q: Express the circle specified by x = 2 cos(0) y =1+2 sin(0) in Cartesian coordinates and find the…

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.