x= sint.cost, y = sin’t 2

Q: 9. The arc of the parabola y = x² from (0,0) to (1,1) is rotated about the x-axis. Find the area of…

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Q: b. The arc of the catenary y= the surface generated. e te from x=a to x=2a, revolves about the…

A: Basic Theory for the above problem:

Q: I. Let C be the parametric curve with parametric equations x=t(t² + 1)-¹/2 te [-3,3] ly: = 2t - 2…

A: There are two points at where the tangent line is horizontal

Q: a) Find the arc length parametrization of the line x = t, y = t that has the same orientation as the…

A: a)The given line is x = t, y = t.We can write it as a vector valued function as r(t)→ =x(t) i^ +…

Q: The parametric curve Osts is rotated about y-axis. Find the surtace area of the obtained ly= | =…

A: Given parametric curve is x=cos(t),  y=sin(t)+t When curve is rotating about y axis, then surface…

Q: ind the arc length parameter along the curve from the point where t=0 by evaluating the integral s =…

A: Given the curve:    r(t)=4etcos ti+4etsin tj-4etk Find s(t), the arc length parameter.

Q: ind the arc length oI the parametric curve. 0 = 6 cos (t), y = 6 sin(t), z = 8t; 0 stST

A: Given: x=6cost, y=6sint, z=8t

Q: Find an arc length parametrization ri(s) of the curve r(1) = (5t, , ), with the parameter s…

A: By bartleby rules, only first question has been answered Given curve is rt=⟨5t,103t32,103t32⟩…

Q: I. Let C be the parametric curve with parametric equations [x = t(t² + 1)-¹/2 te [-3,3] y = 2t - 2…

A: We have to set up the integral that gives the arc length of C.

Q: Find the arclength of the curve 5 cos(2t), y = 5 sin(2t) with 0<t< 8 х Preview

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Q: The given curve is rotated about the y-axis. Find the area of the resulting surface. y = 5 – x2, 0 <…

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Q: Find the area of the surface generated by revolving the curve about the x axis y=√x, 0≤x≤2( Note: I…

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Q: Q7. Consider a large circle of radius R > 0 centered at the origin. Suppose that a small circle of…

A: -------------------------------------------------------------------------------- The circle of…

Q: 39 A parametric curve is given by x =1- y=1+ 0. dy in term of t and evaluate it at 1=-2. dx a. Find…

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Q: Q1 Consider the plane curve defined by the parametric equations x = p + t sin(t),y= p² + t cos(t).…

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Q: Find the arclength of a parametric curve given below for 0 ≤ t ≤ (x(t) = 3 cost (y(t) = 3 sint

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Q: The paraterized curve below is rotated abour the x-axis. Find the area of the surface. x = cos³ (0)…

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Q: 5. For the Parametric curve x = a(t – sint), y = a(1 – cost) where 'a' is a constant. Find y d?y at…

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Q: Consider the parametric equations t3 + 2t, y = e' +t te [1, 10] Find an integral expression for the…

A: We know that area enclosed by x-axis and portion of curve y=f(x) in [a,b] is

Q: OI tr surface obtained by rotating the curve 1 y = 4 1< x < 2, 3x about the x-axis. b). Consider the…

A: a. Consider the surface y=x34+13x, 1≤x≤2. To find the exact area of the surface when rotated about…

Q: 1. Given: parametric curve H defined by r(t) = tan-¹ t and y(t) = e¹, where t € R (a) Find the slope…

A: Solved below.

Q: Find the point on the curve r(t) = (5 sin t)i + (- 5 cos t)j + 12tk at a distance 52a units along…

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Q: se the arc length formula to find the arc length s of the curve y = | vt -ldt , 1<x <4. 1 int: think…

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Q: Find the area of the surface generated by revolving the curve defined by x(t) = 3 cos t, y(t) = 3…

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Q: x = -ln(t + 1); y = sint € [0,1]

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Q: The differential of plane curves described. 1. the cissoids y^2 = x^3 / (a-x)

A: y^2 = x^3 / (a-x)

Q: y= 2. this curve turned around surface. y-aris. Evoluate new Su

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Q: Find an equation of the plane tangent to the following surface at the given point. 13 cos (xyz) =- 2…

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Q: Consider the curve C (a helix) parametrized by r(t) = (cos t, sin t, t), 0<t<4T. a) Compute the…

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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: 1(a) Find the arc length of the curve 2x = 3y2/3 over the interval from y=0 to y=1  1(b) Find the…

A: (a) The arc length of the curve 2x=3y23 over the interval y=0 to y=1 can be calculated by using…

Q: Find all the values of t where the tangent line to the parametric curve is horizontal or vertical.…

A: x=32t-cos tdxdt=32+sin tvertical tangent line:dxdt=032+sin t=0sin t =-32t=2nπ+4π3  and t=2nπ+5π3,…

Q: Find the exact arc length of the curve 2xy = y² + 4 over the interval 2 ≤ y ≤ 4. Find the area of…

A: Given: 2xy=y2+4 over 2≤y≤4 to find the arc length of the curve and given the another equation is…

Q: 11. Finding arc length Find the length of the curve V cos 2t dt, <x</4. 0

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Q: Find the arclength of r(1) - (cost Ar sin r)i+ (sint-t cost)j+3k for %3D

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Q: 4:- Consider the image of the sphere r(0; Ø) = (a cos 0 sin Ø; a sin 0 sin Ø; a cos Ø) where 0 0…

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Q: ey + e-y Find the area of the surface generated by revolving the curve x over 0 < y < In(7) about…

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Q: Find all the values of t where the tangent line to the parametric curve is horizontal or vertical.…

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Q: Find the length of the curve y = In (cos x), 0sxs 4 The length of the curve is (Type an exact…

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Q: The given curve is rotated about the y-axis. Find the area of the resulting surface. y = 5 sxS 21

A: The equation of the given curve is y=13x32.The curve is bounded between x=5 to x=21.We have to find…

Q: Consider the following. x = sin(3t), y = -cos(3t), z = 6t; (0, 1, 2n) Find the equation of the…

A: Given :- The given curve x = sin 3t ,  y =-cos 3t  , z =6t   ;  0,1,2π To determine :- The equation…

Q: Q1// Converting r= 12 cos 0 from polar to the equation of a circle? Q2// Find the resulting volume…

A: Please refer the attached image for complete solution.

Q: 25. A hemisphere has a radius of 3 cm. Draw complete with labels the cross section with equation of…

A: Given radius of hemisphere, R=3 cm Consider a small piece of hemisphere of thickness y. It's volume…

Q: x = In (sect + tant)- sint ';osts ;0<t< about the x- axis 3 b) cost

A: We have to find the area of the surface

Q: A SPACESHIP IS TRAVEIING WITH ACCELERATION act) = , AND HAD INITIAL VELOCITY v (0) = <1,0,0). FI ND…

A: A spaceship is traveling with acceleration a(t)=&lt;et,t,sin2t&gt; At t=0,…

Q: y = 2(1+cos( Ɵ)) Find the area of the surface of revolution formed by rotating the upper half of…

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Q: = (3 cost, 4 sint) for 0 ≤ t ≤. Without doing any Let C be the curve parametrized by r(t):…

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Q: 2. (a) Find the exact surface area that is generated by revolving the portion of the curve y = x²…

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Q: Find equation of (a) the tangent plane to the given surface at the specified point. y = e^x cos (z)…

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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.