Which of the following integrals represents the length of the parametric curve x = t -t, y= t2, 1

Q: 3. Set up an integral to find the surface area generated when the given curve x = where 0<t < 1 is…

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

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Q: Evaluate the unoriented line integral (22 + y° + z²) ds (2x² along the parametric curve a : [0, 1]…

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Q: Evaluate the line integral I of the one-form w = xy dx + 12y dy over the parametric curve a : [0,…

A: Given: One-form ω=xydx+12ydy and the parametric curve α:0, ln2→ℝ2 defined by α(t)=e2t, e-2t. To…

Q: Set up an integral to compute the arc length L of the curve C: r (t) = <글t?, 4t, v2t) from the point…

A: length of arc given below..

Q: Evaluate the line integral y = x² from (-1, 1) to (1, 1) and the line segment from (1, 1) to (−1,…

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Q: Let C be the curve in R³ parameterized by r(t) = (sin t, 2t, cos t) for 0< t< /2. (a) Compute the…

A: See the attachment

Q: express the length of the curve as an integral but do not evaluate it. r = eθ + 1, 0 ≤ θ ≤ π/2.

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Q: Find the area of the closed region enclosed by the curve defined parametrically by x = 3 cos t, y= 2…

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Q: Find the area under the parametric curve x=t(1+t) , y=1/(1+t) ,  between t= [0,1]

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Q: Evaluate the line integral, where C is the given curve. xyeVZ dy, C: x = 4t, y = 3t2, z = 4t³, 0sts…

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Q: Consider the curve defined by the equation xy=25. Set up an integral to find the length of curve…

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Q: Find the area bounded by the parametric curve x=t−(1/t), y=t+(1\t) and the line y=2.5.

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Q: Evaluate the integral (+ 2) dz where C is the line segment from the origin to the point 1+ 2i.

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Q: Evaluate the integral ∫Cydx−xdy over the curve C parameterized by r(t)= (cost,sint), 0≤t≤π2.

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Q: Evaluate the line integral, where C is the given curve. 1 ,C:x =t*, y = t", sis1

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Q: Use a scalar line integral to find the length of the curve r(t) = (20 sin , 20 cos ,5), for 0 < t <…

A: Given

Q: Give the integral representing the length of the parametric curve x=t^3, y=t^4, 0≤t≤1.

A: Given: A parametric curve:   Now the length of the parametric curve is given by the formula:

Q: Consider the curve r(t) = (t, 3 cos t, 3 sin t) for t > 0. COS So r(t)dt Compute the integral above.

A: ∫0πr(t)dt=∫0π&lt;t,3cos t,3sin t&gt;dt               =∫0πtdt i+∫0π3cos tdt j+∫0π3sintdt k…

Q: Evaluate the line integral I of the one-form w = xy dx + 12y dy over the parametric curve a : [0,…

A: Given: ω=xydx+12ydy To find: I=∫αω  where α:[0,ln2]→ℝ2 with α(t)=(e2t,e-2t) ∴x=e2t,y=e-2t Now,…

Q: 2. Set up an integral to find the exact length of the curve defined by 1 f(r) 1 In r %3D where 1<r <…

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Q: Which of the following integrals will find the area of the surface generated by revolving the curve…

A: fx=x3 is a polynomial function. It is continuous in the interval 0,3.

Q: Evaluate the integral |xy dx – z dy + 8z dz where C is the curve with parametric equations x = 2t, y…

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Q: Evaluate the following line integrals in the complex plane by direct integration (not using…

A: Complex

Q: Which of the following integrals determines the length of the curve defined by r(t) = 2ti + cos(t) j…

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Q: Which of the following is the surface area of the surface generated by revolving the curve y = 폴 in…

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Q: Evaluate the area under the parametric curve given by x = 2, y = 4t – 2 and above the x-axis on the…

A: Area of the parametric curves along x+axis

Q: Evaluate the line integral ds along the curve C given by 7()=(2 sint, 2 cost), 0sts T.

A: Given line integral is ∫Cx+12yds. Curve is r⇀t=2sint,2cost,0≤t≤π. xt=2sint,yt=2cost Compute the…

Q: Which of the following integrals represents the length of the parametric curve z = 1+e, y=ť, 3 <t<…

A: In order to know the length of the given parametric curve we will use the formula  where  -----

Q: Set up an integral to compute the arc length L of the curve C: r (t) = (t², 4t, v/2t) from the point…

A: The given curve is: r(t)=12t2, 4t, 2t From point (0, 0, 0) to point 12, 4, 2 the limit of t is:…

Q: Set up an integral to compute the arc length L of the curve C: r (t) = (t, 4t, V2t) %3D from the…

A: It can be found by integrating between limits of t= 0 to 1

Q: 1+y dy

A: We can solve the given integral as follows:

Q: | (xy dx + 2y dy)

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Q: Evaluate the line integral, where C is the given curve. ∫(with c in subscript) x sin y ds, C is the…

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Q: What is the area under the curve given the parametric equatio x = t3 and y = t° over [0,1]? O 1/8 O…

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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: Set up an integral to compute the arc length L of the curve C: r(t) = (, 4t, v2t) %3D from the point…

A: I am going to setup an integral to compute the arc length of the given curve using the required…

Q: The area of the surface obtained by rotating the curve y = -+ X°, 1sx< e about the line x = 24 is x3…

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Q: Evaluate the indicated line integrals in the complex plane by direct integration. Siz dz along the…

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Q: Calculate the integral ∫C(x2+y2)zds, where the curve C is parameterized by r(t)=(sin3t,cos3t,4t),…

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Q: Find the area under the curve for the parametric function defined by the equations x(t) = 5t, y(t) =…

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Q: Set up an integral that represents the length of the curve. = sin²(t), y = 1 –- 3 sin(t), 0 <t < T…

A: The given problem is solved below.

Q: Let the following curve be defined as 7: [0, 2] –→ R³ (3) tH7(t) = tl Evaluate the line integral…

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Q: 3. Set up an integral to find the surface area generated when the given curve x = where 0 <t < 1 is…

A: x=t3,y=t2      o≤t≤1    Given

Q: Use Green's theorem to evaluate the line integral along the given positively oriented curve. | 8y dx…

A: Introduction: Green's theorem is one of calculus' four fundamental theorems, all of which are…

Q: Express the length L of the curve r = sin3(0), 0 < e < as an integral but do not evaluate it. /7 de

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Q: Which one of the following integrals gives the length of the parametric curve O 1. I = S V16t2 +1 dt…

A: Given the following equation of curvex(t)=2t2y(t)=t

Q: 3) Find the area of the surface that results from rotating the curve Y = sin X, 0<X< 1 about the X…

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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…

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.