To illustrate the fact that the numbers we get for length do not depend on the way we parametrize our curves (except for the mild restrictions pre-venting doubling back mentioned earlier), calculate the length of the semicircle y = sqrt(1 - x2) with these two different parametri-zations: a. x = cos 2t, y = sin 2t, 0<=t<=pai/2. b. x = sin pt, y = cos pt, -1/2<=t<=1/2.

Q: A membranc of uniform density area is stracthed on a Circular frame of radius A to uniform stress…

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Q: To illustrate thefact that the numbers we get for length do not depend on the waywe parametrize our…

A:

Q: Given the double intergral 3 S.³ To √²+y² Evaluate the above dy dx intergral using cartesian…

A: some of the formula of integration . ∫1x2+a2dx=ln|x+x2+a2|+C∫xndx=xn+1n+1+C

Q: Show that the volume of solid formed by revolving one loop of the curve r2 = a? cos20 about the line…

A:

Q: The centroid of the half-ellipse-shaped-cross-sectional area shown is given by: 4 Ye 9 - 25m9-y² dy

A: Given half ellipse shaped cross section is:…

Q: What does the arc length formula give for the length of the line y = x between x = 0 and x = a,…

A: line y = x between x = 0 and x = a

Q: 32. Find an arc length parametrization of the curve parametrized by %3D r(1) = (1,/2, 32), with the…

A: A curve traced out by a vector-valued function is parameterized by arc length if. Such a…

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Q: Keparametrize the curve rit) = with respect to its arclength measured from the point (e"20,e"2) in…

A: Given curver(t)=etsint,etcost,et Above curve is passes from the point eπ2,0,eπ2.we need to find the…

Q: A particle is moving in space with position function R(t) = (cos(4t), sin(4t),5 - 3t) %3D Find the…

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Q: the detined b is Constant. Curiature a(t) =| 1 Cost, 1 sint, o of the curve d:laa)tR? where Find

A: We are given : α(t)=1rcos t,1rsin t,0 ⇒α'(t)=-1rsin t,1rcos t,0 Now, the Unit tangent Vector is :…

Q: Consider the ellipse described by x = cost, y= 3sint for te [0,2x]. Construct an integral whose…

A: Answer

Q: Find a parametrization for the following curve. a.) The circle centered at (2,1) of radius 3.

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Q: Compute the length of the circle of radius r defined parametrically by = 3 cost, y = 3 sint ; 0 s t…

A:

Q: The cycloid generated by the unit circle has parametrizationr(t) = (t − sin t, 1 − cost)(a) Find the…

A: Given :           The parametrization of the cycloid is r(t) = (t - sin t, 1 - cos t).

Q: Consider a new curve Č parametrized by r(t) = (cos t esint, sin t- cos t), 0 <t <I but with F still…

A:

Q: For KEIRT and TEIR construct Co-Curve Y: IR→IR³ with constant Curvature K and constant torsion T

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Q: 4. Given YZ tangent to oJ at point Y and m<WYZ = 104, %3D what is the measure of WXY?

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Q: Which of the following is an arc-length parametrization of the line segment from (2,1) to (1,3)?…

A: We will find out the required value.

Q: 4. Find the length of the arc C = {(x, y): y = V16 – x²?,0 < x < 3}.

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Q: USE THE METHOD OF VARIATIOH Of PARAME TERS To SOLUE THE FOLLOWIHG DIFFEREHTIAL EQUA TION!…

A:

Q: What is the formula for the slope dy/dx of a parametrized curve x = ƒ(t), y = g(t)? When does the…

A: Given parametric curves are

Q: Set up the definite integral(s) equal to the arc length of the curved defined by the parametric…

A: Use the formula of arc length in parametric form and simplify.

Q: Q1: Consider the curve in E' parameterized by C:(t) = 1- Find the arc-length of the curve r(t)? 2-…

A: Given equation of the curve is rt=cos5t, sin5t, 12t

Q: Show that the surface area obtained when the arc of the parabola y? = 16az from I = 0 to r = h is…

A: Surface area

Q: Consider the part of the helix ~x(t) = (2 cost, 2 sin t, 3t) that is bounded by t = 0 and t = 2π.…

A: We have to find the arc length of above helix

Q: Consider the parameterization for the cycloid given as: x = 4(1– sint) y = 4(1-cost) 0<t<2n a) Find…

A: Given curve is x=4t−sint and y=41−cost, 0≤t≤2π (a) We have to find the area of the arch of the…

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

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Q: Find the Sufface arca 0f the soid of revolution Obtained bu rotatig y=+2x With TtxE2 around メーaxis、…

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Q: s being pumped into a spherical ballo nge of the radius when the radius reac

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Q: 2. The Cornu (or Euler) spiral, shown below, is defined by r(s) = (f* cos (u²) du, f *cos (u²) du,…

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Q: s) Evaluate the line itegral ( +2y + 2)dy – dz, where C is the arc of parabola z = 2+ y". from (2,0)…

A:

Q: To illustrate thefact that the numbers we get for length do not depend on the waywe parametrize our…

A:

Q: denote the square of the arc length differential over the surface of Z=xy. What do you need to do to…

A: The differential path length is: ds=dx2+dy2+dz2    =dx2+dy2+dxy2    =dx2+dy2+xdy+ydx2

Q: ..... The area of the surface generated by revolving the curve x = 1 cos (2t), y = 7+ -sin (2t) on…

A: Follow 2nd and 3rd step.

Q: From Rogawski ET 2e section 8.1, exercise 13. Approximate the arc length of the curve y = x the…

A: Solve the following

Q: Find the length of the following two-dimensional curve. r(t) = (2 cos t,2 sin t), for 0stsI %3D…

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Q: Find the length of the following three-dimensional curve. r(t) = (6 + 3t,5 – 3t, - 1+2t), for 1sts6…

A: The length of the curve rt=ft,gt,ht in the interval a≤t≤b is given by L=∫abf't2+g't2+h't2dt. The…

Q: Use (a) the Trapezoidal Rule, (b) the Midpoint Rule,and (c) Simpson’s Rule n=10 with to approximate…

A: aGiven∫241lnxdx,n=10Using Trapezoidal…

Q: Show that the total arc length of the ellipse x = 2 cos t, y = sin t (0 <1< 27) is given by /2 4 V1…

A: Arc length is the total distance of the boundary of any curve. If the equation of ellipse is in…

Q: If you compute the surface area of a sphere as the volume of revolution of the parametric curve c(t)…

A:

Q: Find the length of the following two-dimensional curve. r(t) = (17 cos t, 17 sin t), for 0stsx…

A: The given parametric curve is: r(t)=17 cos t, 17 sin t ;         for  0≤t≤π

Q: A flat, very thin table has a circular hole with radius 1 and a circular hole with radius 2. The…

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Q: A bascboll is thrown with an nitial velocity of 45 myr = 6ol6 f%s at an angle morizontal and from…

A: Given: Initial height above the ground h0 = 5 feet Initial velocity v0 = 66 feet/second

Q: a The integral that represents the length of the curve 7 (t) = <2/t,cos (2t3/2), sin (2t3/2)) where…

A: Given curve is, r→(t)=&lt;2t, cos(2t32), sin(2t32)&gt;, t∈[0,1] Now,…

Q: Starting from the point (-3, –1, 1) reparametrize the curve r (t) = (-3+ 3t) i+(-1+2t)j+ (1+ 3t) k…

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Q: Find an arc length parametrization of the line y = 8x + 7. r;(s) II

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Q: By revolving the semicircle x = r cos t, y = r sin t. (0 ≤ t ≤ π) about the x-axis, show that the…

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Q: Find the length of the curve #(t) = et cos(t), y = e* sin(t) for 0 <t < 5. %3D %3D Give an exact…

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Q: The 35-kg boy is sliding down the smooth spiral slide such that his z coordinate decreases 2 m per…

A: Solution is in step 2

Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.