QUESTION 2 Let C be the oriented curve parameterized by r(t)=(t,t> for 0st s2 (so C is actually an oriented line segment). Let F(x,y)= (F1F2) where F1=4x and F2=xy.Compute the vector liné integral dr. QUESTION 3

Q: In words, describe the curve r(t) = (2 sint,e',2 cost) for 0<= t <=6 . Be sure to a -- give enough…

A: To Explain: The parametric curve is r(t)=(2sint,et,2cost) for 0≤t≤6π , so that its graph can be…

Q: 6. Find the unit vectors that are parallel to the tangent line to the curve 2x3 at the point (2,…

A: The given curve is y=2x33 at the point 2,163. Calculate the unit vector that are parallel tot he…

Q: (a) F(t)=(2 sint)i + (2 cost)j from the point A(2, 0) to the point B(-2, 0)

A: To find the arc length of given function

Q: xx+24=5x² fpyonbouy of anyos

A: We have xx+24≤5x2 Consider:…

Q: Find a vector parametric equation F(t) for the line through the points P = (1,0,-5) and Q=(2,3,-7)…

A: The vector parametric equation of the line that passes through the points: A=x1, y1, z1, B=x2, y2,…

Q: Eliminate the and write an equation of- carve in part (a) Sobely in cartesian coordinates x, y (e)…

A: As asked we shall solve only part (d) and (e).We have to eliminate the variable t and write the…

Q: TRUE OR FALSE. A plane curve C defined by the parametric equations x = f(t) and y = g(t), where t is…

A: check the given statement is true or not.

Q: 3. The position vector of a particle moving in the plane is given by r(t) = e *'i+ e'j. a) Find the…

A:

Q: the position vector r describes the path of  an object moving in space. (a) Find the velocity…

A:

Q: Prove that Serret-Frenet formulae can be written in the from t' = dxt, n'=dxn, b' = d x b and…

A:

Q: 10. Given the curve C defined by the vector-valued function r(t) = a) find an equation of the line…

A:

Q: Prove that Serret-Frenet formulae can be written in the from t' = dxt. n'=dxn, h' = d x b and…

A:

Q: Let C be a curve defined by some vector function R with unit binormal B such that R(0) = (-2, 5, 4)…

A: Given C1 is a curve defined by some vector function R→ with unit binormal B→ such that R→(0)=-2,5,4,…

Q: Find T, N, and B for the curve r(t) = (14t, 14t, e') at (0, 0, 1). %3D Hint: After finding T', plug…

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Q: Find Components of the Acceleration For the curve defined by T(t) = (e-t cos(t), e' sin(t)) |3D find…

A: To find- For the curve defined by r→t = e-tcost, et sint, find the unit tangent vector, unit normal…

Q: Write down the vector equations of the tangent lines to the traces for /(r, 3) = re-y² at (2,0) for…

A: To write the vector equation of the tangent line for the below function to the trace y=0.…

Q: Find parametric equations for the line through Po = (10,-1, 1) perpendicular to the plane 2x + 3y -…

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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: