Let F = -2yi + 2xj. Use the flux form of Green's Theorem to compute F·n ds where C is the positively oriented circle x? + y² = 9.

Q: Find the integral surface of the linear PDE xp - yq = z which contains the circle x² + y? = 1, z=1.

A: From Lagrange method

Q: Let C be the positively oriented circle r? + y? =1. Use Green's Theorem to evaluate the line…

A: Just I have used the Green's theorem to change the line integral into double integral and then by…

Q: Let C be the positively oriented circle x2 +y2 =1. Use Green's Theorem to evaluate the line integral…

A: C is the circle  x2 + y2 = 1 The integral to find  ∫c3ydx +9xdy

Q: 10. Let C denote the positively oriented circle z| = 1. Show that - Ti 2sin(z) dz cos(z) dz 4 z(z²…

A: See the attachment.

Q: Let f (z) = SLet C'be the circle |z| = 2in the counter-clockwise direction. What is | f(2) dz 2ri C

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Q: Use Green's Theorem to evaluate pxcosydx – ysinxdy, where the curve C is the square with 0), (TT,…

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Q: Verify greens theorem form 1 for F=x^3i+yj and c is the circle x^2+y^2=1, oriented anti clockwise

A: Find ∫CFdr, the line integral of F, where C is the circle x2+y2=1, oriented anti clock wise…

Q: Evaluate F · dī where C F = , and C is the circle of radius 3 centered at the origin parametrized…

A: To find  ∫CF→.dr→ where c is circle of radius 3 centered  at origin parametrized in counterclockwise…

Q: Sketch the space curve r(t) = ⟨2 sin t, 5t, 2 cos t⟩ and find its length over the given interval [0,…

A: The given equation of space curve is  r(t) = ⟨2 sin t, 5t, 2 cos t⟩.

Q: |Use Green's Theorem to evaluate | -x'ydx+xy°dy where C is the boundary of the C circle x + y =16…

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Q: 21. r(t) = t³ j + t² k %3D

A: Given  r(t)=t3j+t2k

Q: =2) Calculate & ydx + xz² dy - zy³ dz, where c is the circle x² + y² = 4, Z=3 (conterclockwise)…

A: We will use stokes theorem to find integral

Q: Calculate f((5x + 2y)i + (4x + 7y) 3) · dr where C' is the circular path with center (a, b) and…

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Q: Let z be a complex mumber , f() b any funcion, and c be contour any then to f B(Z)dZ?

A: Let z be a complex number, f(z) be any function and c be any contour.     A contour is a type of…

Q: Use Green's Theorem to evaluate the integral (2x? – y)dx + 4xdy where C is the circle x? + y² = 9…

A: To evaluate the given integral using green's Theorem.

Q: Let C be the circle |2- i 2 oriented counterclockwise. Evaluate the following integral 1. z(2² + 4)

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Q: Find F. dr where C is a circle of radius 2 in the plane. x+y+z 9, centered at (4, 3, 2) and oriented…

A: Consider the given equation: And the plane x+y+z=9 centered at 4,3,2.

Q: Find a parametric description for the circle x2+y2-6y=0, oriented clockwise and where t begins at 0.…

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Q: Please evaluate (3y – einl2) dr + (7x + V yt + 1) dy, where C is the circle x² + y² = 9. Hint: use…

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Q: Sketch the plane curve r(t) = ti + t2j and find its length over the given interval [0, 4] .

A: Concept: The calculus helps in understanding the changes between values that are related by a…

Q: Let C be the positively oriented circle x? + y? = 1. Use Green's Theorem to evaluate the line…

A: Given : C is a positively oriented circle x2+y2=1 We have to evaluate the line integral…

Q: Sketch the surface r = 2y? + 32².

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Q: 4. Evaluate the line integral / r y* ds where C is the right half of the circle r² + y² = 16.

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Q: Demonstrate that the circles (x – 2)? + y² = 2 and x? + (y – 2)2 = 2 share tangent at point (1, 1),…

A: Here we will apply the concept of finding the equation of tangent to the two circles at point (1,1).

Q: Let F = (tan-(3x) – y)ỉ + (-2x + sec¬1y)j and C be the boundaries of the circle x? + (y – 4)² = 9 in…

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Q: Verify Green's theorem for $[xy² dx + x²y dy], where y is the circle x? + y² = 4.

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Q: Let C be the positively oriented circle centered at the point zo with radius r>0. Use a…

A: We will first parametize the circle C

Q: 2. Use Green's Theorem to evaluate (7z +y) dy - (r² – 2y) dz where C is are the two circles as shown…

A: theorem:

Q: Evaluate f 3z2dz over C, where C is the upper half circle |z – connects points O and 1+i. -글히 %3D플…

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Q: Use Stoke 's Theorem to evaluate [ V. dr, where v= y²i + xyj + xzk, and c is the bounding curve of…

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Q: dA, where K is the disk r + (y 2 +?

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Q: Consider the following subsets of real plane R² A = {(0,y): ≤ y ≤ 1} B = {(x, y): 0 ≤ x ≤ 1, y =…

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Q: Let C denote the positively oriented boundary of the square whose sides lie along the lines x =+ 2…

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: Let f (z) -Let C'be the circle z 10in the counter-clockwise direction. What is 1 -Lf(2) dz? 2ri

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Q: Let F (tan-(3x) + y)i + (2x + sec-ly)j and C be the boundaries of the %3D circle x? + (y- 4)2 = 9 in…

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Q: Evaluate ffp(4xy + 1) dA, where R is the region in the upper half-plane bounded by the circles a? +…

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Q: Sketch the space curve r(t) = ⟨4t, −cos t, sin t⟩ and find its length over the given interval [0,…

A: Consider the given space curve, rt=4t,-cost,sint use the graphing utility to graph the curve,

Q: Find the upper bound of |[∫c ez^2 + 1 / (z - 1)2 + 5] dz|, where C is the circle |z - 1| = 2.

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Q: Let C' be the circle of equation a² + (y - 2)2 in counterclockwise direction. a) Find the…

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Q: Give an example of a parametrisation of the unit circle, C = {(x,y) E R² |x²+y² = 1}, that generates…

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Q: EVALUATE § 7. d² USING STOKE'S THEOREM 3xz^ + ex² ₁ + 2xy^ с F(x, y, z) = 3XZ1 te C IS THE CIRCLE…

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Q: Let C be the positively oriented circle x² + y² = 1. Use Green's Theorem to evaluate the line…

A: Greens theorem

Q: Let C be the positively oriented circle x² + y² = 1. Use Green's Theorem to evaluate the line…

A:

Q: 5. Let C denote the positively oriented circle z- 1= 3. Evaluate the integral: 3 z +2z dz 3. (z- 2)

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Q: Let C be the curve connecting (1,1) to (-1,-1) clockwise around the circle x2 + y = 2 1 -dy + y? + 1…

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Q: Let C be the positively oriented circle x2 +y2 =1. Use Green's Theorem to evaluate the line integral…

A: The given circle C is  x2 + y2 = 1 Heree the centre of the circle is (0, 0) Radius of the circle r =…

Q: Use Stokes' Theorem to evaluate F• dr where C is oriented counterclockwise as viewed from above. yzi…

A: The given line integral is ∮CF→·dr→ Where, F→=yzi^+6xzj^+exyk^ C is the circle x2+y2=9 at z=8 The…

Q: Evaluate F. dS, where F = 6xy²i + 6x²yj + 2z°k and S is the surface of the unit sphere.

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Q: Evaluate the line integral: x dy – y da x² + y² where C is the circle x² + y? = a², a > 0, oriented…

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Arc Length

Arc length can be thought of as the distance you would travel if you walked along the path of a curve. Arc length is used in a wide range of real applications. We might be interested in knowing how far a rocket travels if it is launched along a parabolic path. Alternatively, if a curve on a map represents a road, we might want to know how far we need to drive to get to our destination. The distance between two points along a curve is known as arc length. 

Line Integral

A line integral is one of the important topics that are discussed in the calculus syllabus. When we have a function that we want to integrate, and we evaluate the function alongside a curve, we define it as a line integral. Evaluation of a function along a curve is very important in mathematics. Usually, by a line integral, we compute the area of the function along the curve. This integral is also known as curvilinear, curve, or path integral in short. If line integrals are to be calculated in the complex plane, then the term contour integral can be used as well.     

Triple Integral

Examples: