Find parametric equations for the circle x2 + y2 = a2, using as parameter the arc length s measured counterclockwise from the point (a, 0) to the point (x, y).

Q: Find the length of the arc of the curve having the parametric equations x = t³ and y = 2t2 from t =…

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Q: Find an arc length parametrization of the line y = 2x – 6 %3D |

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Q: Identify the length of the arc having parametric equations x = t3 and y = 2t2 from t = -2 to t = 0

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Q: Find the arc length for the parametric x = e' cost; y = e' sin t; 0sts2n and sketch it curve below.

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Q: A circular helix in xyz-space has the following parametric equations, where θ ∈ R. x(θ) = 4 cos θ…

A: Arc length in 3d We want to determine the  arc length of a  curve given by parametric equation ,…

Q: Give a parametric description for a cone with radius a and heighth, including the intervals for the…

A: We can parameterise the same as following: Assume s & t as two parameters radius = a and height…

Q: Determine the arc length parameterization of the circle x^2+y^2 =a^2 with orientation…

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Q: find the areas of surfaces generated by revolving the curves about the given axes

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Q: Find the arc-length of the segment of the curve parametric

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Q: Find the arc length of y3 + y = x2 + 6x on the interval [ -5, -1 ]. Also, determine the surface of…

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Q: 4) Find the curvature of y = cos x at the point (÷,²). Then find the equation of the osculating…

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Q: Find the arc length of the curve y 1 on the interval [1, 2]. %3D 4x2

A: Given:    y = x4/8 + 1/4x2   Interval : [ 1 , 2 ]

Q: The parametric curve X= 2sin(t) + 2, y=3cos(t) – 2 describes a clockwise traveling vertical ellipse…

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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…

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Q: Find the arc length form (2,1) to (4,4) of the curve x = 2/y.

A: The formula for arc length is L=∫abf'x2+1dy.

Q: Express the following curves parametrically, including the interval for the parameter. 2.) 2.) A…

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Q: The unit circle x² + y=1 can be wrriten in parametric form as x = cos(0) and y : = sino. dy and…

A: Follow the procedure given below

Q: For which values of t is the parametric curve x=8t, y=t+t, -0c0s ts 00 concave up? (Enter your…

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Q: The parametric curve x=2sin(t)+2, y=3cos(t)-2 describes a clockwise traveling  vertical ellipse with…

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Q: Find an arc length parameterization of the circle in the plane z = 9 with radius 4 and center (1, 4,…

A: The parametrization of the circle in the plane z=9 is:                                   r(t)=⟨1, 4,…

Q: Find the arc length (8) of the curve 7 (t) = ( – 2e*, – 4e*, – 2e*) fo %3D c - - -

A: Explanation and solutions is given below....

Q: Find the arc length function for the curve y = 2x3/2 starting at the point (1,2). %3D

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Q: Find the length of the parametric y = 2 , - V3sts V3. curve x = 1+

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Q: Obtain the center of gravity of the skin of the rotating object that occurs if the arc y = √9 - x2;…

A: As per the question we are given the arc : y = √(9-x2)  for  0≤x≤3 And the surface obtained by…

Q: Find the arc length of y³+y=x²+6x on the interval [-5,-1].

A: Solution:   Given that,      y3+y=x2+6x,  interval -5, -1         Then, differentiating with…

Q: Find the arc length of the curve y = 2/3x 3/2 from (1, 2/3) to (2, 4/ 3).

A: "Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Find the length of the arc of the curve having the parametric equations x = t³ and y = 2t2 from t =…

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Q: Which type of surface does the curvex= cos(3t),y= sin(3t),z=t, lie on?

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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: Determine the surface of revolution if the arc y^3 + y = x^2 + 6x on the interval [−5, −1] is…

A: The curve will appear like this

Q: Suppose a curve is defined by the parametric equations x=sin(t) and y=cos(t) i) Find the points…

A: Explanation of the answer is as follows

Q: Find parametric equations for the semicircle x2 + y2 = a2, y > 0, using as parameter the slope t…

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Q: Find the length of the arc of the curve having the parametric equations x = from t = 0 to t = 1. t…

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Q: The area of the rotational surface resulting from rotating x² + y2 = a² about the x axis is? the arc…

A: Given : x2 + y2 = a2

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

A: given that x=3 cost , y=3 sint; 0≤t≤2π find out the length of the circle of radius r i.e. arc length…

Q: Find the length of the circle of radius r defined parametrically by x=rcost, y=rsint,0≤t≤2π.

A: Here I am attaching image so that you understand each and every step.

Q: Find the equation of the tangent plane at the point (-2,1 – 3) to the ellipsoid surface + y? + = 3.

A: Find the equations of the tangent plane and normal line at the point (-2, 1, -3) to the ellipsoid…

Q: Evaluate integral and use parameter t to show parametric equations for circle ∫c x6y d s where C…

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Q: Find arc length for the curve (y+1)2=4x3 x=0 ,x=1

A: This question can be solved using the concept of length of arc of a given curve over a given…

Q: Find parametric equations for the semicircle x2 + y2 = a2, y> 0, using as parameter the slope t =…

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Q: Show that the arc length of the graph of y = sin x on the interval [0, 2π] is equal to the…

A: Given, Equation of Ellipse is x2+2y2=2 Now divide the equation of ellipse by "2" we get,…

Q: Find parametric equations for the arc of a circle of radius 3 from P = (0,0) to Q = (6,0). x(t) =…

A: Given :-   A circle of radius 3 from P=(0,0) to Q=(6,0)  To find:-  Find parametric equations for…

Q: Give an equation of the surface of revolution generated by revolving x In z about the z-axis.

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Q: Find the arc length (in parametric form) over the interval: 0stsi

A: The arclength can be obtained by using the expression L=∫t1t2dxdt2+dydt2dt. Here, dxdt is the…

Q: 2) Find the cartesian eauation for the olar curve r=tan e sece ×ニy )yニxと 2.

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Q: The curve y = V4-x², -1<x<1, is an arc of the circle x2 + y² = 4. Find the area of the surface…

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Q: Find the arc length of y^3+y=x^2+6x on the interval [-5,-1]. Using the same arc, determine the…

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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.