Calculate the integral of f(x, y, z) = 3x² + 3y2 + z4 over the curve r(t) = (cos(t), sin(t), t) for 0 sts n. %3D %3D = ل )م تود تم( 3x2 + 3y2 + z) ds %3D

Q: Find an equation of the normal line to the parametric surface given by R(u, v) = ((2+ cos v) cos TU,…

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Q: Evaluate the line integral (2 yi + (r- 2)j+ zyzk) · dr where C is the parametric curve r(t) = ti +…

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Q: QI) Evaluate the integral dy dx using polar coordinate

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Q: Q1) Evaluate the integral dy dx using polar coordinate

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Q: B) r(t) = (4 cos t, 4 sin t, 16)

A: We consider x=rcos(theta) and  y=rsin(theta)  Therefore r^2=x^2+y^2

Q: Find (d^2)y/d(x^2) at t= pi/3 for the parametric curve given below. Write your final answer in the…

A: Given parametric curve, x = t+sin(t) y = t-cos(t)

Q: Find the total distance traveled by a particle along the curve given in parametric form x=sint -t…

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Q: Q1) Evaluate the integral dy dx using polar coordinate

A: Introduction: Multiple integral can e converted from one coordinate system to another coordinate…

Q: Find the slope of the tangent line to the parametric curve 2te x(t) y(t) at the point P(0, 1). 2t +e…

A: xt=2t etyt=2t+etAt  P 0,1,  x=0,   y=1  ∴x=0⇒2t et=0             ⇒t=0   dxdt=ddt2t et= 2 t…

Q: | (xy dx + 2y dy)

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Q: Let C be the curve connecting (0,0,

A: Given that C be the curve connecting  (0, 0, 0) to (1, 4, 1) to (3, 6, 2) to (2, 2, 1) to (0, 0, 0).…

Q: 4. Consider the parametric curve x = cost, y = sin cost, te [-T, π]. (a) Show that (0,0) is a point…

A: Consider the parametric curve given by  and , where  belongs to .a) To show that  is a point of…

Q: Find the parametric equations of normal line to the surface given by f(x, y) = tan v x2 +y2 at the…

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Q: Does f(z)=z/(sin z)^2 have a pole of order 1 or 2 at z=0?

A: f(z)=zsin2zFor the poles first we look at the zeros of the denominatorat z=0 We know sin(z) has 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: