Use Green's theorem to evaluate $ 2y3 dx – x3y² dy, where C is the positively oriented circle of radius 2 centered at the origin.

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: Question 2. Use Green's Theorem to evaluate f, 2y?dx + 5xydy, where C is the boundary of the…

A: We have to evaluate the integral ∫C2y2dx+5xydy where C is the region between the circles x2+y2=1 and…

Q: B) Use Green's Theorem to evaluate /F-dr, where C is the circle r² + y? = 4 and F (r, y) = (y – 2…

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Q: find the areas of surfaces generated by revolving the curves about the given axes

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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: Calculate the line integral ∫Cy2ds, where C is a part of the circle x=acost, y=asint, 0≤t≤π/2.

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Q: Let C be the curve of intersection of the parabolic cylinder x² = 2y, and the surface 3z = xy. Find…

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Q: The curve C is the circle of radius 5 contained in the plane z 5 and centered around the z-axis,…

A: Solution:-

Q: Let r < R. Revolving a circle of radius r, (x – R)² + y² = r², around the y-axis gives a donut…

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Q: Q2 Calculate 8- z dz, z(4- z) where C is the circle of radius 7, centre 0, negatively oriented.

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Q: Q2 Calculate 8- z dz, z(4-z)* where C is the circle of radius 7, centre 0, negatively oriented.

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

A: Given: C is a positively oriented circle x2+y2=1.  Objective of the question is to calculate…

Q: Find an arc length parametrization of the circle in the plane z = 11 with radius 4 and center (10,…

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Q: Q2 Calculate 8-z dz, z(4-z) where C is the circle of radius 7, centre 0, negatively oriented.

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Q: Use Green's Theorem to evaluate the integral (3x² – y)dx + 3xdy where C is the circle a? + y? = 1…

A: As we know from the Green's theorem; ∫CMdx+Ndy=∫∫D∂N∂x-∂M∂ydA   Given; I=∫C3x2-ydx+3xdy So, M=3x2-y…

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: 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: Use Green's Theorem to evaluate the integral (x² - y)dax + 3xdy where C is the circle x² + y² = 1…

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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: 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: 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: Suppose a point X lies in the exterior of a disk, and suppose that two lines are drawn from X…

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Q: Evaluate the line integral ſe ry²ds, where C is three quarters of the circle a² +y? = 1, oriented…

A: C is the three quarters of the circle x2+y2=1, oriented counterclockwise. The parametric equation of…

Q: 6. Compute the contour integral (a) / COS Z -dz where C is the positively oriented circle |z – 1| =…

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Q: Find a parametrization c(θ) of the unit circle such that c(0) = (−1, 0).

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Q: Prove that the curve given by the equations x=R(cost+tsint),y=R(sint−tcost) is the involute of the…

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Q: The tangent plane to the graph of the surface z=In(x2+y2) at the point (1,0,0) contains point…

A: Given function is, z=lnx2+y2

Q: Use Green's Theorem to evaluate the line integr y? dx + x • when C is the circle of radius 2…

A: Introduction: Let C be a positively oriented, piecewise smooth, simple, closed curve, and D be the…

Q: Evaluate the exact area of the surface obtained by rotating the circle x2 + y² = r² about the line y…

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Q: 9. Using Stoke's Theorem Evaluate [(2x – y)dx – yz dy – y zdz) where c is the circle x +y =1,…

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Q: 157. Calculate -x²ydx + xy? dy, where C is a circle of radius 2 centered at the origin and oriented…

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Q: Jse Green's theorem to evaluate $ (2°e* + 4y) dz + (-5x+ y* sin(y)) dy, where C is the…

A: The given path C is circle x2+y2=16. Therefore: D= (x, y) : x2+y2≤16

Q: Consider the curve C which is the intersection of the surfaces shown in the attached figure.

A: To evaluate the below integral. ∫Cx−y−2zds

Q: Use Green's Theorem to evaluate - – In(x² + y²))dx + 2 tan¬²(y/x) dy, C is the circle (x - 4)² + (y…

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Q: KS is tangent to the circle at T. secant KR contains P, the center of the circle. PR and PT are…

A: Consider the given information. Let's draw the graph step by step given as KS is tangent to the…

Q: Use Green's Theorem to evaluate (1in z – 2y) dz + (x – ycos y) dy, where C is the circle (x – 4)² +…

A: Given: C is the circle x-42+y-42=9. ∮Cxsinx-2ydx+x-ycosydy. To find: Using Green's theorem, value…

Q: 4. Let C be the positively oriented circle centered at the origin with radius r> 3. Without…

A: Given: Let C be a positive oriented circle centered at the origin with radius r&gt;3. Without…

Q: Find the centroid of the arc of the circle r = 2sin0 + 4cos0 from 0 = 0 to 0 = T1/ %3D O (2.637,…

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Q: sin z 0 dz, where C is the positively oriented circle of radius 1, |z| = 1. %3D

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Q: Evaluate the contour integral and use appropriate graphs to illustr where is the circle with…

A: Given : ∮e2021zz2+1z2-1dz To Find : (a) z-i =1                                                     b…

Q: Let C denote the positively oriented circle z-i=2. Evaluate the integral: 1 dz +4

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Q: Evaluate ds where C is the upper half of the circle centered at the origin with radius 1 with the…

A: Draw the curve C:  Here, we divide C in two parts:  C1 represented by the curved part C2…

Q: Find the curvature of the parabola y = x² at the origin.

A: Given :-   parabola y=f(x) = x2  To find:-  curvature of parabola at origin(0,0)

Q: Calculate both sides of Green's Theorem when F = [-x²y, xy²] and y positively oriented circle x? +…

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Q: the line integral SexY² ds where C is the of the Circle x² + y²=4 oriented clockwise Evaluate Cight…

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Q: Evaluate the double integral. (8x – 6y) dA, D is bounded by the circle with center the origin and…

A: The given data is: ∫∫D8x-6ydA To evaluate the double integral.

Q: The tangent plane to the graph of the surface z=In(x2+y²) at the point (1,0,0) contains point…

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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: