Evaluate the line integral, where C is the given curve. xy ds,     C: x = t2,    y = 2t,    0 ≤ t ≤ 5

Q: Find the points on the curve a = t – 16t + 1, y = t² + 12t – 8 that have: %3D A. a horizontal…

A:

Q: Ja² xz ds, where C₁ is the curve defined by Ŕ(t) (Int, 3-4√t, 2t), te [1, 2]. Evaluate

A: Given:  Curve C1 defined by R→t=lnt,3-4t,2t , t∈1,2. To find: ∫C1xzds.

Q: Let C be the curve represented by the equations 5t, y = t, (0 < t< 1). In each part, evaluate the…

A:

Q: Find 2 for the curve dx x²y³ – 2xy = 6x² + 2y3 + 1

A: Topic = Derivative

Q: 2. Let F = (-y/2, x/2), and let C be the curve shown below. What is F ds? Explain %3D why. y 1 1 2 3

A: The vector field is F=-y2, x2 C1:0, 0→2, 0rt=2t, 0, r't=2, 0 F=0, t Hence, the integral becomes…

Q: Find the tangent to y = cotx at x = - T O A. y = - 2x + 2 π +1 +1 OB. y=2x-- OC. y=2x+1 OD. y=-2x +…

A:

Q: Evaluate the line integral, where C is the given curve. | xyevz dy, C: x = 2t, y = 4t2, z = 213,…

A: Consider Applying derivative with respect to t,

Q: Select the statement that describes the curve xy =c in quadrant I, if c is some positive constant…

A: To select the correct statement that describes the curve  in quadrant I, if is some…

Q: Let C be the curve represented by the equations x = 10t, y = t², (0 ≤ t ≤ 1). In each part, evaluate…

A:

Q: cost, y = sint, at the point where t 0 ≤t≤ 2π -O?! =

A:

Q: Let C be the curve represented by the equations X = 11t, y = t², (0 ≤ t ≤ 1). In each part, evaluate…

A:

Q: Let C be the curve represented by the equations X = 4t, y = t², (0 ≤ t ≤ 1). In each part, evaluate…

A:

Q: Find the tangent X = 3t² + t Y = 4+³ 3 line t = 1 to the curve

A:

Q: /

A:

Q: Compute f ds for the curve specified. f(x, y) = V1 + 9xy, y = x3 for 0 < x < 2.

A:

Q: Find the length of the curve y = V- from z =1 to r = e. 3

A:

Q: Find the points on the curve x = £² – 4t − 6, y = t² + 10€ that have: : A. a horizontal tangent line…

A: Let us consider the given curve x(t),y(t). The horizontal tangent line is obtained as dydt=0 The…

Q: Let F = (4x³ + yz, xz, xy). Let C be any smooth curve from (1, 1, 1) to (2, 3, 0) Calculate cF dr.

A:

Q: Let's Check Activity 1: Determine the equations of the tangent to the curve defined by: 1. f(x) = x³…

A:

Q: y = sec ² t 2 y=1-₁² 0 t

A:

Q: Evaluate + 5x)dx + (ye* – 3y²)dy where C is the boundary of the curve given below. -2

A: Detailed explanation mentioned below

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: