6. Let C C R? be the positively-oriented boundary of the graph of the circle x2 + y? = 1. Using techniques learned in this course, evaluate 2²y dx - ry dy. (6.1) Solution to Problem 6.

Q: - Compute 5y dr a dy, where C is the arc of the parabola y = from (-2, 4) to (0, 0).

A: The line integral of the function with respect to x, y is solved using the formula…

Q: 3. What is the length of the curve r = cos3 (-) where 0 < 0 <: sos

A: See the attachment

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

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Q: 2-3 2P

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Q: 5. Use sheees to tirel the udure di the s reuolutian: Reçim bameled loy about Hhe axes : Y2x ) メ-0…

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Q: Let C be the circle of radius 3 given by r(t) = 3 cos ti + 3 sin tj, 0 st s 2n as shown in Figure…

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Q: 4. Compute, ydx + xdy x² + y² where C is the circle centered at the origin of radius 2 oriented…

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Q: 5. Let C be the circle x +y} =4_which is positively oriented. Let C, be the upper half-circle C1 =…

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Q: Find the length L of the arc formed by y=e^4x/4,0≤x≤3

A: To find the length L of the arc formed by y=e^4x/4,0≤x≤3

Q: 10. If the curves f(x, y) = 0, y, z) = 0 touch, show that at the point of contact, dr dø dr dø dy dx…

A: According to our guidelines, we can answer only the first question, the rest can be reposted.

Q: 4. Let C be the circle [z| = 4 traversed once in the positive sense %3D sin 3z (counter clockwise).…

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Q: 2: Evaluate O (6y+x)dx+(y+2x)dy where C is the circle (x–2)²+(y-3)² = 4 | a) Using Green's Theorem

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Q: 8., (x + y) dA, where R is the region that lies to the left of the y-axis between the circles x? +…

A: As you have asked to answer question 8, we will answer question 8.Question 8.The given integral…

Q: Find the arc length of the graph of y^2=4x^3, between points (0, 0) and (2, 4sqrt(2)).

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Q: 16 6xy ds, where C is the right half of the circle x² + y² = 4. Evaluate the line integral

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Q: 19. Evaluate Scx ds, where C is a. the straight-line segment x = t, y = t/2, from (0, 0) to (4, 2).…

A: Part (a)

Q: Problem 5. Compute the positively oriented contour integral e32 dz 2² + 4

A: ∫z=5 e3zz2+4Here we use residue and  solve this question

Q: 15. When a line L is tangent to the circle 2 + v = 100 at the point (a, 6), find the equation of…

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Q: Let S CR³ be the portion of the graph of the cone z2 = x² + y? which satisfies 0 s z < 1. Using…

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Q: Consider the region 2 enclosed by the curves I', and I': given by: T: y=1 Ti y=(x- 1)²

A: Given curves are, Γ1:y=1Γ2:y=x-12 So, the required diagram of the given curves are,

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

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Q: 12. Use Greens Theorem to calculate P y'dx -a'dy C where C consists of the arcs y=x² and y =x1/2,…

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Q: 2 Green's theorem, evaluate ((+) dx + 3xy dy where C is the circle * +y = 4.

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Q: 3 The arc length of the portion of the curve y=x 2 with 0<x<4 is

A: I have append the formula used to find arc length in the solution part.

Q: Evaluate O (6y+x)dx+(y+2x)dy where C is the circle (x-2)²+(y-3)² = 4 Without using Green's Theorem.

A: We will find out the required value.

Q: Compute . Z+1 dz, where C1 is a part of a circle with r = 3, and 0<0< %3D

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Q: A circle, with area A = tr“, is expanding with time. dA = 6 cm 2/s, what is dt dr when r = 2? dt If…

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Q: 17. Evaluate OF . • dr where C is the circle x +y² = 1, z = 0 and F=yi+zj +xk.

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Q: Draw a sketch of the given arc C and evaluate S [(x² −2y)dx + (2x + y²)dy]. 2. C is the line segment…

A: Given integral: ∫Cx2-2ydx+2x+y2dy C is a line segment from  12,1 to 54,0 54,0 to 54,2   To find:…

Q: ) Let C be the arc from the point (3,0) to the point (0,3) on the circle x² + y² = 9. The value of…

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Q: 4. Let C be the the circle ² + y² = 4 and Calculate the work(W = [,F dr when C is traversed in the…

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Q: II. Draw a sketch of the given arc C and evaluate Sc [(x² − 2y)dx + (2x + y²)dy]. → 1. C is part of…

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Q: 4. Evaluate / (z – 2) dz where C is the lower half of the circle |2 – 2| = 4.

A: Note: According to bartleby guidelines we have to answer only first question please upload the…

Q: Find the length of the tangent to the circle x² + y² − 2x − 3y − 1 = 0 from the point (2, 5)

A: Solution- The equation of the circle x2+y2-2x-3y-1=0 The point =2,5 length of the tangent

Q: : The area of the surface obtained by rotating the curve y = 2 In a, for 1 < a <e, about the a-axis,…

A: We have to find the integral that given the surface area.

Q: Find the length of the curve over the given interval. r = 2a cos θ, [−π /4, π/4]

A: Since, we have to find the length of the curve over the given intervalr=2a cosθ over the interval…

Q: 9. Find the area of the surface generated when the given arc is revolved about the x-axis: 4x² +…

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Q: 9/ Find the Vale of the integrel ) Losz+Retz)r Imtz)} dz 3 is the Straitht line Segment from Zero to…

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Q: 29. Let C be the quarter of the unit circle in the first quadrant, traversed coun- terclockwise from…

A: Explained below

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: 6. Compute G.T ds if K is the portion of the circle x2 + y* = 7 in the third quadrant (oriented…

A: region of integration is part of circle, x2+y2=7, lying in 3rd quadrant, thus, limits of x,y are -7…

Q: 3. Discuss how to evaluate f () dz , where C is a simple closed curve C=10 enclosing the origin.

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

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Q: in R°sketch the graph of the cylinder x = (y – 2)?

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Q: dz b) C: z|= 4 is a circle calculate the integral .(z-1)(z² +9)

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Q: Use Green's theorem to find the k number that satisfies (k + 3) ∫C((k/2)*(x^k)*(y^2))dx + ((yx^(k +…

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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: et +e- find the length of the curve h(x) = from r = 0 to z = In 2.

A: The standard formula for arclength will be used in solving this problem. The typeset solution is…

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

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