Find the area of the surface swept out by revolving the para- metric curve x = 5 cost + 5t sin t y = 5 sint – 5t cos t about the x-axis.

Q: the surface area of the surface generated by revolving the curve c(t) = (t, sin(t)) about the x-axis…

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Q: Write an integral for the area of the surface generated by revolving the curve y = cos(4x) about the…

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Q: The curve y= sinh x, 0< x <9 is revolved around the x-axis. Find the area of the resulting surface.…

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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: Find the area of the surface that is obtained by rotating the curve Y = 0.25X? – 0.5 Log.X, 1<X< 2…

A: Given curve is, y=0.25x2-0.5logex  1≤x≤2 Let the area of the curve be S.

Q: Find the area of the surface generated by revolving the curve x= cos (3t), y = 5+ sin (3t) on 0sts,…

A: Suppose parametric curve rt=xt,yt for a≤t≤b are revolving about the x-axis then surface area…

Q: Find the moment of inertia with respect to x- axis of the area between y = sin x from x = 0 and x =…

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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 of the surface generated by revolving the curve a. y = V1+ e* between x = 0 and x = 1…

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Q: 1. Find the moment of inertia, with respect to the line x=1, of the area enclosed by the loop of the…

A: Given: Curve y2=x2(1-x)----(i)

Q: Find the surface area of the region that comes from rotating the given curve about the x-axis: T X =…

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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: The area under the curve parametrized by x(t) = e", y(t) = 2et +1, where 0 St<1/3, is equal to.

A: As given, x(t) = e3t  y(t) = 2e3t + 1 where 0≤t≤13.

Q: Find the area of surface revolution, when x=1+cosθ. , y=sinθ Where 0≤θ≤2π

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Q: Calculate the area of the surface formed by rotating the curve r = 2a cos 0 around the line 0 = pi/2

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Q: Find the area of the surface generated by revolving the C = ((x, y): x = 3 (t – sin t), y = 3 (1-…

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Q: Find the mass of a thin wire shaped in the form of a helix, r(t) = ( 3 cos t, 3 sin t, 4t) with 0 <…

A: The curve is given by  rt=⟨3cost,3sint,4t⟩ , 0≤t≤3π2                   (i) Then,…

Q: Find the area of the surface generated by revolving the curve a. y = V1+ e* between x = 0 O and x =…

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Q: Find the surface area of the figure generated by revolving x = 6 cos(t), y = 6 sin(t) where 0 <ts…

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Q: Find the area of the surface generated by revolving the curve x = a cos θ,       y = b sin θ, 0 ≤ θ…

A: The curves x=acosθ,y=asinθ, and 0≤θ≤2π, we have to find the area of the surface generated by…

Q: Find the surface area of the object obtained by rotating y = sin(2x) , 0<x< about the x-axis.

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Q: Find a curve through the origin in the xy- plane whose length is given by 1/2 L = [; (1+ *)"* dx 4

A: The arc length, L of a curve y=f(x) in the interval a,b is calculated using the formula below.…

Q: . Use the integral table and a calculator to find to two decimal places the area of the surface…

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Q: ) Find the area of the surface that is obtained by rotating the curve Y = 0.25X² – 0.5 Log.X, 1<X<2…

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Q: determine the area of the curve r = v V3 cos e + sin 0 that is below the r-axis.

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Q: Evaluate the line integral (x+ 1 ds along the curve C given by r (t) = (2 sint, 2 cost), 0<t<n %3D

A: Given: rt=2sint, 2cost We know that rt=x,y So, x=2sint, y=2cost ⇒dxdt=2cost, dydt=-2sint Also, we…

Q: )Find the volume of the solid obtained by rotating the sine function between x 0 and x t about the…

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Q: Find an arc length parametrization of the line y = 8x. r:(s) = %3D

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Q: 36. Find the area of the surface generated by revolving the para- metric curve x = cos² 1, y = sin?…

A: Definition used -                               Area of the surface generated by revolving the…

Q: Find the area of the surface generated by revolving the given curve about the indicated axis. 1. x =…

A: Consider two parametric equations and with . Then the area of the surface obtained by rotating…

Q: Determine the moment of inertia of the shaded area about the x-axis. 2" 10" y Answer: Ix Parabolic =…

A: To determine the moment of inertia about the x axis.

Q: Find the area of the surface generated by revolving about x-axis the part of the curve y = sin x,…

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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: Find the center of gravity of the region bounded by the x-axis, the curve y = sec x, and the lines x…

A: Consider the given question,The graph of the curve y = sec(x) and the line x = -pi/4, x = -pi/4 is…

Q: Consider the pavametenzed cunve given by X=cos, y= sin (t), 0ste2 TT a) Sketch the curve b) set up…

A: SOLUTION-

Q: The area bounded by the ourves y= *", and y=2x iş revolved about the y-axis, Find the moment of…

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

A: Given: xt=-2cost  , yt=3sint , 0≤t≤π2 Formula used: The area under a parametric curve: A=∫abytx'tdt

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: find the area of the surface formed by revolving the cardioid r= a(1+cosθ) about the x-axis

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Q: Find the arc length of the parametric curve: x = e'(sint + cos t), y = e'(cost - sint) OstsIn3

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Q: Use any method to find the Area of domain bounded by curve parametrized by X = cos³t₁ Y = sin³t, 0≤…

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Q: Calculate the area between the curves r2 16 sin 20 and p2 16 cos 20 .Show by drawing. %3D %3D

A: We will draw region first, then find the area using double integral

Q: Find the area of the surface generated when the indicated arc is revolved about the y -axis. 6. y =…

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Q: b) Sketch the graph and find the points on the cardioid r=a(1+cos) at which tangent lines are…

A: Consider the equation of the cardioid, r=a1+cosθ The graph will be,

Q: Find the area of the surface of revolution obtained by rotating the function y=sin(x) on the…

A: Here is the given curve and the interval it is bounded by: y=sin(x) interval {0,pi} about the x…

Q: How many points are there on the surface z = dx +y such that the tangent plane to the surface is…

A: Given :  d=maxa,b,c⇒d=max2,1,0⇒d=2 So,z=dx2+y2⇒z=2x2+y2 Also, dx+y+z=0⇒2x+y+z=0 Let fx,…

Q: Set up and evaluate the definite integral for the area of the surface generated by revolving the…

A: Consider the given integral as  2π∫-5581-x2dx

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.