Evaluate the line integral directly and by using Green's Theorem where C is positively oriented.

Q: Express the length of the curve as an integral but do not evaluate it. r = sin θ cos θ, 0 ≤ θ ≤ π

A: We have givem expression r=sinθcosθ, 0≤θ≤π We have to set up the integral for calculate the length…

Q: Assuming that the line integrals in the options are over the unit circle centered at the origin (x2…

A: .

Q: The contour of integral zdz of the cirle |z-i| =2 is

A:

Q: Use Green's Theorem to evaluate the line integral

A:

Q: Evaluate the integral or show that it is divergent.

A:

Q: Use the Fundamental Theorem of Line Integrals to calculate exactly. F-6 sin(ox +)+ sin(6x )j along…

A:

Q: Evaluate the integral using FTC I.

A:

Q: Evaluate the integral using Part 1 of the Fundamental Theorem of Calculus. 5 4x(1 – x² )dx = i

A:

Q: Use the Fundamental Theorem of Calculus to evaluate (if it exists) -5t- dt. If the integral does not…

A:

Q: show the line integral is independent of path

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Q: Use the Fundamental Theorem of Calculus to decide if the definite integral exists and either…

A:

Q: Evaluate the integral. ∫t3  et dt

A: ∫t3  et dt  We need to evaluate the integral.

Q: Make a substitution to express the integrand as arational function and then evaluate the integral.

A: Consider the integration as

Q: Define f : [0, 1]→R by f(x) = x. •1 Use the definition of the integral to prove that 1 x dx |

A: Definite integral: An integral that contains the upper and lower limits then it is a definite…

Q: Сompute 1. f dx, where f(x)=|x|.

A:

Q: Assuming that the line integrals in the options are over the unit circle centered at the origin…

A:

Q: Use the fundamental theorem of calculus part 2 to evaluate the 3 integral: (8 sin x) dx. Noto: Round…

A: Our Aim is to evaluate the integral given below:-∫π/4π/3 (8 sin x) dx -(i) using Fundamental Theorem…

Q: Evaluate the double integral (x +y)² sin(x – y) dA for R, which is square with the vertices (0, 1),…

A: Consider the given integral and construct a diagram based on the given conditions.

Q: evaluate the integral (Usec for the constant af mtegration) (Xt 6x) cos (x) dx

A: Use the integration by parts to simplify.

Q: Use the Fundamental Theorem of Calculus to evaluate (if it exists) 4t-5 dt. If the integral does not…

A:

Q: Evaluate the improper intergal 1 dx using the residue (x2 + 9)* theorem.

A:

Q: Evaluate the line integral F.dr

A: In this question, we find the line integral for given vector function and parametric path.

Q: F(x) = sin(t) dt %3D 3x

A:

Q: Evaluate the integral (1+ etane) Sec?0 de

A: The given integral is, I=∫0π41+etanθsec2θdθ Let us solve the given definite integral using the…

Q: Evaluate the integral in terms of u. sinh u• du = + C %3D 3.

A:

Q: Compute the line integral of the scalar function f(x,y) = sqrt(1 + 9xy) over the curve y = x^3 for…

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Q: Verify that the Fundamental Theorem for line integrals can be used to evaluate the following line…

A:

Q: Given

A:

Q: Compute the line integral of F along the oriented smooth curve C shown below. (2,0, 2) (1,4, 3)

A: F =(ysinz)i +(xsinz)j+( xycosz)k dr =(dx)i +(dy)j+(dz)k line integralx-21=y-0-4=z-2-1=t (x,y,z)…

Q: sin 4zdz J 16+ sin ª 2= 4.

A: The given function is sin 4z16+sin4 2z We have to evaluate the integral of this function.

Q: Evaluate the line integral, where C is the given curve.

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Q: express the length of the curve as an integral but do not evaluate it. r = sin3 θ, 0≤ θ ≤ 2π

A:

Q: Evaluate the integral S de using the two-point Gaussian quadrature rule

A: For the integral I=∫abf(x)dx, the Gaussian quadrature 2-point formula is,I=b-a2w1fm1+w2fx2where…

Q: Evaluate the double integral where D is the triangle bounded by the x-axis, y-axis and the line y =…

A:

Q: a) by parametrizing C, b) by using the Fundamental Theorem of Line Integrals.

A:

Q: | 7x2 V2x3 - 1 dx

A: Here the given integration I=∫7x22x3-1dxNow Substitute as2x3-1 =u⇒6x2dx=duThus the integration…

Q: TRUE OR FALSE? The given line integral is independant of path.

A: The given line integral is ∫C2xz3+6ydx+6x-2yzdy+3x2z2-y2dz Compare the above line integral with…

Q: Evaluate the integral; f, Ni-1, tsin(at)) dt

A: Evaluate the definite integrals  in the following  way:-

Q: Compute the integral using contour integration: 27 cos no (cos 0+a)20 for a > 1.

A: Note:- ∫C fz dz =2πi ×sum of residues cosnθ= Re ei nθ  cosθ=ei θ+e-i θ2      Given that  ∫02π…

Q: Use Green's Formula to evaluate the line integral along the positively oriented closed curve C:

A:

Q: Make a substitution to express the integrand as arational function and then evaluate the integral.

A: Consider the given integral ∫133xx2+xdx Divide the numerator and denominator by x, get…

Q: State the Fundamental Theorem for Line Integrals

A: To State: Fundamental Theorem for Line Integrals Here we will use the knowledge of fundamental…

Q: (a) using a parametric description of C and evaluating the integral directly, and (b) using the…

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Q: Write the integrand in terms of u. 2x (x + 1)' dx = du %3D Evaluate the integral.

A:

Q: Is the integral / A · dr path independent if A = (2x - y)i+ (x + y)j? of Evaluate A · dr around a…

A:

Q: Evaluate / F. dr using the Fundamental Theorem of Line Integrals.

A: In this question, we used the fundamental theorem of vector calculus. here first we prove the vector…

Q: Use the Fundamental Theorem of Calculus to decide if the definite integral exists and either…

A: The solution are next step

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: