(b) Find eiz dz- (z + i)3 Izl=1.5 (the circle is oriented counterclockwise).

Q: Find the moving trihedral and the curvature at any point of R(t) = (t²,– cos 8t, sin 8t).

A: Moving trihedral are three unit vectors of curve  Unit tangent vector  Unit normal vector  binomial…

Q: 2.2 Prove that Te -dz < 3 ez z2 +7+1 where C is the arc of the circle |2| = 3 from z = 3i to z = -3.

A:

Q: Given S as a quadric with equation 4x^2 - (y-3)^2 + 4z^2 = 4 Find a generating curve for S (as a…

A: Given S as a quadric with equation 4x^2 - (y-3)^2 + 4z^2 = 4 Find a generating curve for S (as a…

Q: A torus of radius 2 (and cross-sectional radius 1) can be represented parametrically by the function…

A: The surface area (A) of torus is given by the formula mentioned below with given values as.

Q: Evaluate COS Z -dz , where C is the circle |z – 1| = 3. (Integrate counterclockwise around C.)

A:

Q: 32. At what points does the helix r(t) = (sin t, cos t, t) intersect the sphere x? + y² + z² = 5?

A:

Q: Suppose that a cylindrical container of radius r and height L is filled with a liquid with volume V…

A: Consider the given:  Express volume in terms of “h”, “r” and “w”.…

Q: For the following problem, a point is rotating with uniform circular motion on a circle of radius r.…

A: We know that angular velocity is

Q: Find the equation of the plane perpendicular to the curve r(t) = (sin(2t), t, cos(2t)) at the point…

A:

Q: Chapter 12, Section 12.3, Question 006 Find the arc length of the parametric curve. x = 6 cos t, y =…

A:

Q: Line integrals Evaluate the following line integral using a method of your choice.

A: We will use Green's Theorem to evaluate the integral , according to which we have

Q: Consider the plane curve given by the parametric equations x = t² + 3t - 26 y = t² + 3t - 28 - What…

A:

Q: 3. What is the length of the curve r = cos3 (-) where 0 < 0 <- 4

A:

Q: te the line integral c 3 == 1 from A=(2, 0) to (0, -1) Jax + 2xyay where is the arc on the ellipse

A:

Q: Find the arclength of y = 6x – 3 on 0 < x < 4. Give the exact answer. -

A:

Q: 2. KS is tangent to the circle at T. Secant KR contains P, the center of the circle. PR and PT are…

A: Firstly draw a circle with center P Now draw a tangent KS at point T

Q: Evaluate where C is the portion of the circle centered at the origin with radius 2 in the 1st…

A:

Q: Compute sin z dz , where a is the piece of the parabola with equation y= a?, which lies between the…

A:

Q: A slime mold grows in a spiral pattern with polar radius r=θ3r=θ3. If the mold grows by continuing…

A:

Q: 32. (sin y, x) • dr, where C is the boundary of the triangle with vertices (0, 0), 0 ), and 2' 2

A:

Q: Sketch the plane curve r(t) = a cos3 ti + a sin3 tj and find its length over the given interval [0,…

A: The curve is r(t)=a cos3ti + a sin3tj   Interval is 0, 2π

Q: 2.2 Prove that e? dz] < 3 +++1 where C is the arc of the circle |2| = 3 from z = 3i to z = -3.

A:

Q: 3. Find T (t) and a set of parametric equations for the tangent line to the helix given by ř(t) = 2…

A:

Q: Find the moving trihedral and the curvature at any point of R(t) = (3t2, sin 2t, cos 2t).

A: Introduction: The Frenet–Serret formulas in differential geometry describe the kinematic properties…

Q: The unit circle x² + y=1 can be wrriten in parametric form as x = cos(0) and y : = sino. dy and…

A: Follow the procedure given below

Q: Consider the curve C parametrized byx = −1 + 6 sin t and y =(1/2)− 6 cos t for −π ≤ t ≤ 7π. What is…

A: First of all, consider the given parametrized equation of curve C. Rearrange the equations to write…

Q: Find the speed of the cycloid c(t) = (4t −4 sin t, 4−4 cos t) at points where the tangent line is…

A:

Q: Consider the curves:r = 4sin 0 and r = 2 sin(0) + 2cos (0).

A:

Q: 1. Find the moving trihedral and the curvature at any point of the curve R(t) = cos t î + sint ĵ+ k.

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: 4. Find the parametric equation of the tangent line to r(t) = 4costi – 3tj %3D at Po(2, –n)

A: Given: To find the parametric equation of the given tangent line as follows, r(t) = 4 costi- 3tj…

Q: 4.2. Calculate the arc length of the cardioid given by r = a(1 – cos 0) -

A: The cardioid is symmetrical in nature and ranges from θ=0 to θ=π as a half figure. We can also…

Q: Find the exact value of arc length of the semicubical parabola y2 = x³ between the points (1,1) and…

A:

Q: Find the equation of the osculating plane of the helix: r(t) = sin(3t), y(t) = 2t, 2(t) = cos(3t)…

A: Given the equation of the helix:     x(t)=sin(3t), y(t)=2t, z(t)=cos(3t) So, rt=sin3t,2t,cos3t. Find…

Q: 3. Evaluate S, 2 – xy ds where C is the upper half of the circle centered at the origin of radius 1…

A: The general form of the parametric equation of the curve in two-dimensional space is given by x=f(t)…

Q: Compute ž dz +y1 z dz, +72 and where +Y1 is the segment from 1 to -1, and +y2 is the arc on the unit…

A:

Q: Page 9 9. The curve y = x2 with 0Sx<1 is rotated about the y-axis. Find the exact area of the…

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:

Q: 2.2 Prove that e zp- 3 where C is the arc of the circle |2| = 3 from z = 3i to z = -3.

A:

Q: ez 16. Evaluate dz, where: (4z-1) cos(Tz) a) I is the circle |z| = 1/8 traversed once…

A: We need to evaluate the complex integral in the following three regions represented by Γ,

Q: 6) Find the equation of the osculating plane of the helix x = sin 3t, y = 2t, at the point (-1,л,0).…

A: Given :x=sin3ty=2tz=cos3tpoint -1, π, 0

Q: Prove that e dz < ++1 3 where C is the arc of the circle |2| = 3 from z = 3i to z = -3.

A:

Q: Find the moving trihedral and the curvature at any point of R(t) = (t²,– cos 8t, sin 8t). -

A: Given is : R→(t)=t2,-cos(8t),sin(8t) In order to find the curvature use the formula . Formula for…

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

A:

Q: A curve is defined by the parametric equations z (t) = at and y (t) = bt, where a and b are…

A: Given,  We have  xt = at   and yt = bt  ;  where a and b are constants.  We have to find the length…

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

A:

Q: 4. Consider the circular helix given by 7(t) = (3 cos t, 3 sin t,4t). Give an equation for the…

A:

Q: (b) cos(z) -dz, z2 + 2iz – 1 where y is the circle {|z + i| = 1} with counter clockwise orientation.

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: