A curve C is defined by the parametric equations x = t² + t – 1 y = t - t? a) Find in terms of t. dx b) Find an equation of the tangent line to C at the point where t = 2

Q: Evaluate the arc length of the curve generated by the parametric equations below for t (-1,1) x(t)=…

A: The expression for the arc length of the parametric curve is; L=∫abx't2+y't2dt   Given: xt=3+3t2 ;…

Q: Find the length of the curve r = cos(f) + sin(0), 0sOs

A: The objective is to find the length of the curve.

Q: Consider the parametric curve given by (a) Find dy/dx and d²y/dx² in terms of t. dy/dx = |d²y/dx² =…

A:

Q: Sketch the path using the following parametric equations x(t) = 3t – 7 and y(t) = 4 sin t+1 0st55,…

A: Here we need to find different points for different values of t

Q: Let C be the curve with parametrization r(t) = cos(t)i + sin(t)j – 2tk. a) Find the parametric…

A: Let C be the curve with parametrization   rt=costi+sintj-2tk We have to find parametric equations of…

Q: Consider the parametric equations x(t) = 1 + 3t2, y(t) = 4 + 21³, -1 <t <1. • What is the length of…

A:

Q: A curve is defined by the parametric equations z (t) = e-* and y (t) = e*. What is in terms of t ? A…

A:

Q: A curve is defined by the parametric equations: x(1) = e' cos t, y(t) = e' sint d²y dy and dr of the…

A: It is given that, x(t)=et cost,y(t)=et sint   It is known that, dydx can be found by chain rule and…

Q: the length of the parametric curve a =t+cos(t), y =t – sin(t), 0 < t < 2n ?

A: The length of the curve formula is

Q: Consider the parametric curve C that is defined by C: x=t^3, y=t^2+t, −1≤t≤3. (a). Find the…

A:

Q: A curve is defined by the parametric equations x (t) = at² and d² y y (t) = bt where a and b are…

A:

Q: Sketch the curve defined by the parametric equations x = sin πt/2, y = t, 0≤ t ≤ 6.

A:

Q: A curve is defined by the parametric equations sin z dz, CoS Z dz, t 2 1. y= X= Find the length of…

A:

Q: Consider the curve C in R³ defined by the parametric equation c(t) V2 v2 T = (cos(t), sin(t), t).…

A:

Q: Consider the parametric curve defined by 1 a(t) %3D y(t) = t² || What is the slope of the line…

A:

Q: A curve is defined by the parametric equations x (t) = e¬t and y (t) = e3t. What is d'y in terms of…

A: The solution for the above problem is shown below.

Q: Find the length of the curve defined by the parametric equations t, y = 5 In((t/6)? – 1) I = from t…

A: Given, x = 56t , y = 5lnt62-1 = 5lnt2-36-5ln(36)and 7≤t≤10

Q: The curve defined by any parametric equations X = f(t), y = g(t) is always a function.

A:

Q: Without eliminating the parameter t, find a Cartesian equation for the lin curve C, defined by the…

A:

Q: Find an equation of the tangent(s) to the curve with parametric equations tsint at the point…

A:

Q: Find the exact value of x²y³z ds, where C is the curve with parametric equations x = e-t cos(5t), y…

A:

Q: Consider the curve r(t) = (t, 3 cos t, 3 sin t) for t 2 0. Find an equation for the projection of…

A: Equation of projection of curve on xy plane will not contain the term of z.

Q: Find the length of the parametric curve x = e^t + e^-t, y = 5-2t on the interval [0,3]

A: Find the length of the parametric curve x = e^t + e^-t, y = 5-2t on the interval [0,3].

Q: A curve C is defined by the parametric equations x(t) = 3 + t“ and y(t) = t° + 5t.Which of the…

A: Given,         A curve C is defined by the parametric equations xt=3+t2 and yt=t3+5t.

Q: Let C1 be the curve with parametric equations x = cos t, y = sin t – t, where t E [0, 5). (a) Find…

A: This question can be solved using the concept of equation of tangent and length of curve when the…

Q: A curve C has parametric equations X = tチ+1. dy (i) Find in terms oft. dx (ii) Show that there are…

A: We first find dy/dx

Q: Find T(t) and then find a set of parametric equations for the tangent line to the helix given by…

A: Consider the equation of helix given by r(t). Find the value for t , derivative at t and modulus of…

Q: Consider the curve r(t) = (t, 3 cos t, 3 sin t) for t 2 0. Find an equation for the projection of…

A:

Q: Let C1 be the curve with parametric equations x = Cos t, y = sin t – t, where t e [0, ). (a) Find…

A:

Q: Sketch the plane curve represented by the parametric equations x = 6 cos t, y = 4 sin t, π ≤ t ≤ 2π…

A: Given  x = 6 cos t, y = 4 sin t, π ≤ t ≤ 2π

Q: Let C1 be the curve with parametric equations x = cost, y = sint – t, where t e [0, 5]. (a) Find the…

A:

Q: In which direction is the curve x = -2 sin t, y = 2 cos t, for 0 ≤ t &lt; 2π, generated?

A: x = -2 sin t, y = 2 cos t Table for values of x and y: t x y 0 0 2 π 0 -2 π2 2 0 2π 0 2

Q: a) Find the are length of the parametric curve x = B(t. sint), y = B(1+ cost), in the range 0 sts,…

A:

Q: Find parametric equations for the line tangent to the helix r(t) =(sqrt(2)cos t)i + (sqrt(2) sin t)j…

A:

Q: Find the length of the parametric curve I = cos(t) y = 1 + sin(t) from t = 0 to t =

A: Given- x=cost and y=1+sint To Find- The length of the parametric curve from t=0 to t=π. Identity…

Q: Show that the curve with parametric equations x = t cos t, y = t sin t, z =t lies on the cone z? =…

A: We have to check if x,y,z parametric equations lies on the given cone.

Q: Find the equation of the tangent line(s) to the given set of parametric equations x = t² + 5t – 6 y…

A:

Q: Consider the curve C parametrized by(x, y) = (5 cos (−12t), 5 sin(−12t)), for −π/2 ≤ t ≤ π/2 What is…

A:

Q: Calculate the length of the curve γ (t) = (cos(2t), sin(2t)) t ∈ [0, π].

A:

Q: Consider the parametric curve C: x=t³ + 1, y = t² – t,0 <t < 3. Find the equation of the tangent(s)…

A:

Q: Find an equation of the tangent line of the parametric curve, I = 1+÷, y=t³ –t² +3,

A:

Q: A curve is described parametrically as x = e' sint, y = e'cost, – o <t< ao. Find the equation of the…

A:

Q: Find the length of the curve C defined parametrically by C: r(t) = y(t) = e' sin t, 0<t<n e cost,…

A:

Q: Find T(t) and a set of parametric equations for the tangent line to the helix given by 7(t) = 2…

A:

Q: Which of the following is the equation of tangent line of the curve that is given by parametric…

A:

Q: Find the length of the parametric curve defined by x=r°, y=t- for 0st<1 7.

A:

Q: Find the tangent line equation to the parametric curve cos t y = sin t at t = O A. y = x+ O B. y =…

A: If the parametric equations of a curve are given variable t, then the slope of the tangent is the…

Q: Sketch the curve given by the parametric equations x= t³ - 1 and y= t² + t.

A:

Q: Find the length of the curve C defined parametrically by C:x(t) = e' cos t, y(t) = e' sint, 0<tS#…

A:

Q: Given the parametric curve x(t) = 3t + In(t), y(t) = 3t – In(t)

A:

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: