II -3 -3 -3 II 3 IV 3 11 -3 3. 3.

Q: find the domain of the vector-valued function.

A: Given: rt=1t+1i+t2j-3tk

Q: Using || || and 0, give the vector in (x, y)-form. II

A: We have to write the x,y form of vector v.

Q: The acceleration of an object after t seconds is given by the vector-valued function a(t) . | If…

A: to find the velocity equation, take integration  .............(1)

Q: Find the vector equation for the line passing through the points P₁(-3, 6, 8) and P₂(−6, 0, 12). 0 0…

A:

Q: Find the vectors direction of F= -2yi-xJ of the points below :- (-12-1), (-21-1),(3,0). (-1₁-2),…

A: Vector, F→=-2y i^-x j^Points: -1, -1, -2, -1, 3, 0, -1, -2, -3, -2, -1, 0

Q: X = Find the vector equation for the line of intersection of the planes x − 5y + 5z = −5 and x + 5z…

A: Given the equation of planes x-5y+5z=-5 and x+5z=1. To find the vector equation for the line of…

Q: A vector equation for the line through the points A (3, -4, -3) and B (3,4,2) is (x, y, z) = . (You…

A: Concept to be used:If a line passes through two points A,Bwith position vectors aand brespectively,…

Q: Let L7 be the line passing through the points Q1=(3,-5, –1) and Q2=(0, 4, –10). Find a value of k so…

A: Given that L1 be the passing through the points Q1=(3,-5,-1) and Q2=(0,4,-10). And the line  L2 be…

Q: Represent the line segment from P to Q by a vector-valued function. P(4, 0, 7), Q(3, -3, 4) r(t) =…

A:

Q: A particle moves at constant speed 1445 units along the curve of intersection of the two the two…

A:

Q: Find the maximum rate of change of f at the given point and the direction in which it occurs. f(x,…

A:

Q: In vector form, the equation of a line is given by r = (0, 1, 2) + t(2, 1,0), where t e R. T…

A:

Q: Let L₁ be the line passing through the points Q₁-(-1, -3, 1) and Q₂=(-4, -6, 2). Find a value of k…

A:

Q: i. BC if x = 2 and y =2

A: In right angle triangle BOC.

Q: Find a vector equation for the line through the point P = (-3, 5, -4) and parallel to the vector v =…

A:

Q: Describe the graph of the equation. r = (6 – 21) i+ 10rj O It is a line in 2-space through the point…

A: Solution :-

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: