Using the formula, Area = " y(t)x'(t)dt, find the area under the following parametric curve. x = 2 sin(t),y = 3 cos²(t) for 0 sts

Q: Find the area of the surface obtained by rotating the given curve about the x-axis. - 10 10 cos (0),…

A: Find the derivatives with respect to θ

Q: A. Find the area enclosed by each of the following curves. T= 4 sin e

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Q: Find the area under the curve given by parametric equations below, above the x axis for 0≤t≤2n (x(t)…

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Q: Set up, but do not evaluate, an integral for the length of the curve. y = 6 cos x, Osxs 27 dx

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Q: Consider the curve defined by y = + from x = 6x! 1 to x = 3. The length of this curve is L = Si…

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Q: Find the area of the region that lies inside the curve r = 2 + cos (2a), but outside the curve r = 2…

A: The given curves are r = 2 + cos (2a), r = 2 + sin(a). Use online graphing calculator and sketch the…

Q: Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the + 7) dx, n = 4 5 Exact…

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

Q: The value of the integral / (t tan-+3y+sin-r)dr + (cot-'y+e +7x)dy, where C is r +y =9 is %3D

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Q: Find the area enclosed by the given parametric curve and the y-axis. sin²(t), y = 8 cos(t) X = y 0.2…

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Q: Find the area under the curve for the parametric function defined by the equations x(t) = -2 cos t,…

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Q: Find the area under one arch of the cycloid x = a(t - sin t), y = a(1 - cos t).

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Q: Which of the following integrals represent the arc length of the curve y cosx ,0<xSn (a) 1+sin (x)…

A: The formula for the arc length is given by: L=∫ab1+(f'(x))2dx...(1)

Q: Find the arclength of the curve in the given interval. x = 3 cos? t , y= 3 cost sin t , 0 s<t <t

A: solution is written in below steps.

Q: Which integral represents the length of the curve y = cos x be- tween 0 and ? I V1 + cos? x dx, I…

A: the given equation of the curve is: y=cosx we have to determine which integral represents the length…

Q: Find the area under one arch of the cycloid. x = 6a(t- sin t), y = 6a(1 – cos t)

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Q: Evaluate the integral S x? dx using trigonometric substitution. V16-x2 Note: Use C for an arbitrary…

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Q: Find the area of the region that's inside the curve r=cos2(Θ) and outsid the curve r=1+sin(Θ)

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Q: Set up an integral that represents the length of the part of the parametric curve shown in the…

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Q: Find the value of integral | (x + y + z)ds, where C is parmeterized by 7(t) = (2/2 cos( – t), 3 sin(…

A: We will solve this line integral

Q: a. Polar coordinates Show, by changing to polar coordinates, that ra sin B In (x² + y²) dx dy = a²ß…

A: (a) Consider the given integral. ∫0asinβ∫ycotβa2-y2lnx2+y2dxdy=a2βlna-12 Where, a&gt;0 and…

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

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Q: 5 sin t (where t e R) is A, then what is A? %3D If the area enclosed by the curve x = 5 cos³ t and y…

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Q: Evaluate the integral |2x sin y dy dx. 0 0 The value is (Type an exact answer, using a as needed.)

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Q: What's the area under the curve of y= cos(t)/sin(t) between t=0 and T=π/2

A: We integrate the equation to find the area at given points.

Q: Which of the following integrals represent the arc length of the curve y = sint ,0 S xS (a) 1+sin2…

A: We have to find the arc length and in second question we have to identify the graph of given…

Q: Reparametrize the curve with respect to arc length measured from the point where t = 0 in the…

A: Here we have to reparameterize the curve with respect to arc length.   I have appended the formula…

Q: Which of the following integrals represent the arc length of the curve y = cosx ,0< x< n (a) V1 +…

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Q: Write the integral(s) you would use to find the arclength of the curve X = t - sin (t), y = 1 –…

A: We have the equation of the curve : x=t-sin(t) y=1-cos(t) ,0≤t≤2π We have to write the integral that…

Q: Find the area enclosed by the given parametric curve and the x-axis. x = 2 sin(t),  y = sin(t)…

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Q: The area between the two curves in * the figure below is y = cos x y = sin 2x T/6 T/2

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Q: Sketch the region and find its area. - The region between y = sin z and y = sin 2r, for 0< I<T.

A: Given: The regions y=sin x and y=sin 2x Consider the two curves are equal, sin x =sin 2x  sin 2x -…

Q: r=l+ cos 0 r= cos 6

A: We will answer the first question as we don't answer multiple question at a time. Please resubmit…

Q: Find the area of the surface generated by revolving the curves about the indicated axis. 5) x = sin…

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Q: Which of the following integrals represents the length of the parametric curve x =t+ cos(t), y =t -…

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Q: The area between the two curves in the figure below is y = cos x y = sin 2x T/6 T/2

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Q: Use a familiar formula from geometry to find the area of the region described and then confirm using…

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Q: Use one of the formulas below to find the area under one arch of the cycloid x = t - sin(t), y = 1 -…

A: Given arch of cycloid, x=t-sin(t), y=1-cos(t)

Q: For this problem, consider the polar curve r= 2 + sin(20), 0 s0ST/2. %3D (a) Find the area enclosed…

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Q: JT 9 cos' t and y 9 sin't (where t E R) is A- then what is 8. If the area enclosed by the curve x =…

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Q: Use a familiar formula from geometry to find the length of the curve described and then confirm…

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Q: Find the area of the region common to the interiors of the cardioids r=1+ sin 0 and r=1- cos 0. A-…

A: we have given r=1+sinθ and r=1-cosθpoint of intersectionis…

Q: Find the area enclosed by the given parametric curve and the x-axis. x = 4 sin(t), y = sin(t)…

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Q: Find the exact area between a large loop and the enclosed small loop of the curve r = 1 + 2 cos(30).…

A: (1).    The given curve is: r=1+2cos3θ

Q: 36. x = sin t, y= sin t cos t, 0<t<T/2

A: Here x=sint and y=sintcost where 0≤t≤π2we have to find the area enclosed by parametric…

Q: Find the area of the closed region enclosed by the curve defined parametrically by a = 4 cos t, y =…

A: We use application of integration.

Q: Find the arc length of the given curve on the specified interval. This problem may make use of the…

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Q: Find the area under the parametric curve x= 2(t + cos t), y=2(1 - sin t) from t=0 to t= pi.

A: We have to find the area under the parametric curves whose equations are : x=2t+cos ty=21-sin t…

Q: Use one of the formulas below to find the area under one arch of the cycloid x = t - sin(t), y = 1 -…

A: Area under one arch of the cycloid is

Q: 1 Simplify the integral dx using an appropriate trigonometric substitution. V(a – 1)2 – 1 sec 0 de…

A: Given- ∫1x-12-1dx To Simplify- The above integral.

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.