If F = V(4x? + y*), find fc F dr where C is the quarter of the circle x2 + y? = 9 in the first quadrant, oriented counterclockwise. ScF- di =

Q: Find the tangent line to the given circle as directed. x² + y² + 10x – 6y – 2 = 0 parallel to y =…

A: Equation of the circlex2 + y2 +10x -6y -2 = 0⇒x2 + y2 + 10x -6y -2 + 25 - 25 + 9 -9 = 0           .…

Q: Set up an intergal of the Curve for the length 3T Y%3DZE* sin (x), O4 xE 2

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Q: Evaluate x+y JF dx dy x² + y² + a² over the portion of the first quadrant lying inside the circle x²…

A: Answer is given using polar coordinates.

Q: Let F(x, y) = 4yi + (2æ + ev* ); Calculate ,F · dr, where C is the circle of radius 2 centered at…

A: Given that F(x,y)=4yi+2x+ey2j The objective is to calculate ∫CF.dr where C is the circle of radius 2…

Q: Show that [(x- y) dx + (x – y)' dy] = 3ra" taken along the circle x +y =a° in C the counter…

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Q: T. 9. TO = PO 10.Sector of a circle is the region bounded by two radii and their intercepted arc.

A: here we check whether the given statement is true or false.

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: If C'is the circle of radius 3 centered at the origin, starting at the point (3,0) and moving around…

A: Answer and explanation is given below..

Q: In circle X, XY = 2 and mZY XZ = 160°. Find the area of shaded sector. Express your answer as a…

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Q: Evaluate (3y - esinz)da + (7x+ V +1)dy where C is the unit circle

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Q: dz The value of the integral Jc 22+9 where c is the circle z - 3i| = 4 is %3D

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Q: Find the length of the arc of x²+y²=64 from x=-1 to x=-3, in the second quadrant.

A: Given curve: x2+y2=64

Q: Set up an intergral for the length of the curve, 04X y%3DZE* sin cx), O4 XE

A: Explanation of the answer is as follows

Q: Find f (x-y3)d a + xy dy if the contour C looks as follows: It consists of two arcs of radius 1 and…

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Q: Re(z) dz, where C is the right half circle |2| = 3, from z 2² – 2i (Ъ) -3i to z = 3i.

A: To Determine :-  An upper bound of the absolute value of the integral. ∫C Re zz2 - 2i dz  where C is…

Q: Find the length of the spiral r = eθ/√2 from θ = 0 to θ = π.

A: Given Differentiating r with respect to thita ,we get

Q: Calculate $ -a? y dæ + xy dy, where C C is the circle of radius 2 centered on the origin. Select…

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Q: In circle R with MZQRS = 74 and QR = 17 units find area of sector QRS. Round %3D %3D to the nearest…

A: Given that ∠QRS = 74 degree QR = 17 To find the area of the sector QRS.

Q: Find the arc length of the graph by partitioning the x-axis. y=(1-x2/3)3/2 , from x= 1/64 to x=1

A: Given function is y = y= 1-x2/33/2 from x= 1/64 to x=1 The arc length of a curve on the interval [a,…

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: Find f (x-y3)d a + xy dy if the contour C looks as follows: It consists of two arcs of radius 1 and…

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Q: The diagram shows an initial arm of length1 unit being rotated countercockwise about the origin to…

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Q: If C is a circle of radius 7 centered at the point ( 4, -2 ), then evaluate $ (7y – esin (2) ) dæ +…

A: Given:  C is a circle of radius 7 centered at the point (4,-2).

Q: 4. Find the length of the arc C = {(x, y): y = V16 – x²?,0 < x < 3}.

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Q: 4. Evaluate y ds where C is the upper half of circle x² + y² = 9, traversed in the counterclockwise…

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Q: Find . F.dr when F(x,y) = (2x,3y²) and C is only the first quadrant part of a circle of radius 3…

A: Let C be a smooth curve and r→t=xti^+ytj^+ztk^ be the position vector of any point x,y,z on C.…

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

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Q: Evaluate o y dx – x³ dy, where C includes the two circles of radius 2 and radius 1 centered at the…

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Q: If C is a circle of radius 9 centered at the point ( -5, 5 ), then evaluate $ (2y – esin ( x ) ) dæ…

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

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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: Given ZY = 25, What is the distance from ZS %3D U W T. S Y

A: Given:

Q: Find the length of the spiral r = O for 0 < 0 < 4. %3D Answer :

A: Given : Polar function for a spiral r = θ and interval 0≤θ≤4   To find : Length of the spiral for…

Q: Find the area of the region in the second quadrant that lies between the circle r = π and the spiral…

A: Second quadrant is  [pi/2 ,pi]

Q: Integrate counterclockwise around the unit circle (|z| = 1) and choose from the options below that…

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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: Find | xdy + ydx, where C is the arc of the curve y = eª for 0 < x < 3

A: Given Data: The given integrand is: ∫Cxdy+ydx The equation of the curve is: y=ex ......1 The limit…

Q: 5. Evaluate (y +tan x)dx+(x -2ry+y)dy, where C is the circler +y' =4 and is oriented in a positive…

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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: Find the arc length of the graph by partitioning the x-axis. y=1/3 (x2 +2)3/2 , from  x = 3 to x =…

A: Given: The function y=13x2+232  fromx=3 to x=4

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: 8. PT is tangent to circle O at P If mNP = 90°, and mMXP : 186°, find %3D %3D X/ d. mZ4 e. mZ5 f.…

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Q: Find the arclength on y = 4x3/2 from x is greater than or equal to 1 and less than or equal to 3

A: given: the function is y=4x32. we have to find the arc length on y=4x32 from x is greater than or…

Q: rea of the closed region det

A: Given, the equation of the circle is                                        (x+2)2+(y-3)2=4

Q: Find an equation for the tangant line to the circle x2 +y² = 2 at the point (1,1 (hint: use the…

A: The slope of the tangent at any point on a curve is equal to the first derivative of the function at…

Q: Suppose Vf(x, y) = 3y sin(xy)i + 3x sin(ry)j, F = Vf(x,y), and C is the segment of the parabola y =…

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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: $ ydx – xdy where C is the circle of radius 3 with center at (13, 28)

A: We use Green's theorem to evaluate the given line integral

Q: Find the areas of the regions Inside the oval limaçon r = 4 + 2 cos θ

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Q: If C is the circle of radius 3 centered at the origin, starting at the point (3, 0) and moving…

A: -81π/2

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: