If C is a circle of radius 8 centered at the point ( 0, 5 ), then evaluate pain(2dr + (11æ sin(y* +g}}dy. 7y sin(y + y))dy.

Q: Find the area bounded by the trigonometric curves Ⓒ 3.450 square units O 2.828 square units O 1.414…

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Q: Let P(cos t, sin t) be a point on the unit circle x2 + y2 = 1 in the first quadrant. Show that t is…

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Q: 32. At what points does the helix r(t) = (sin t, cos t, t) intersect the sphere x? + y² + z² = 5?

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Q: What does the arc length formula give for the length of the line y = x between x = 0 and x = a,…

A: line y = x between x = 0 and x = a

Q: Use Green's theorem to evaluate 2y3 dx – x³y2 dy, where C is the positively oriented circle of…

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Q: The curve C is the circle of radius 5 contained in the plane z 5 and centered around the z-axis,…

A: Solution:-

Q: 2. Evaluateſ. a+ (z+30:) dz along C which is the circle |z + 3i| = 3, counterclockwise. %3D (z+3i)

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Q: Find the arclength of y = sin (x) on the interval [0, a]. %3D

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Q: O show mat me Nomal point @ X= a( Coro +o Smo) y= a(8in0-o Cor0) dy Sine %3D ep" of Normal S-ゴ」=…

A: the parametric function is x=a(cosθ+θsinθ) y=asinθ-θcosθ Differentiate each with respect to theta…

Q: Find the length of arc of the curve y = In r from r = 3 to r= 9. 8

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Q: 11. Evaluate 2 dA, where D is the shaded part inside the semicircle of radius 2 in Figure 26. 2…

A: To Find:  D is the shaded part inside the semicircle of radius 2 From the figure, the restrictions…

Q: 3 Use Green's theorem to evaluate [x³y dx - xy²dy, where C C is the unit circle x²+y=1 with…

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Q: Using Green's theorem, evaluate f y°dx – x³dy, where C is the positively oriented circle of radius 4…

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Q: Express the circle specified by x = 2 cos(0) y = 1+ 2 sin(0) in Cartesian coordinates and find the…

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Q: 1. Use Green's Theorem to evaluate |F. dr, where F(r, y) = (3y- esin, 7x + Vy + 1) and C is the…

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Q: If C is a circle of radius 3 centered at the point ( 3, -4 ), then evaluate £(3y – ein(=) ) dz +…

A: The integral is given by ∮C3y-esinxdx+10x-siny3+ydy Where, C is a circle of radius 3 centered at the…

Q: If C is a circle of radius 7 centered at the point ( 4, -2 ), then evaluate $ (7y – esin (2) ) dæ +…

A: Given:  C is a circle of radius 7 centered at the point (4,-2).

Q: Given the cardioid r= 3(1+cosθ) 0 ≤ θ ≤ 2π Determine in wich points the cardiod has vertical…

A: Differentiate the equations r=31+cosθ and x=rcosθ with respect to θ, to find drdθ and dxdθ.…

Q: Find the speed of the cycloid c(t) = (4t − 4 sin t, 4 − 4 cost) atpoints where the tangent line is…

A: Given: c(t)=4t-4sin(t), 4-4cos(t)

Q: Ex: Fmd the lengthoftre arc af the Circle with P.R. ret)= [acest from Co,0 to asiut] o Ofa),- %3D

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Q: The curve y V4 – x², – 1 < x < 1, is an arc of the circle x² + y? = 4Find the area of the = 1…

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Q: Compute the arc length of the indicated portion of the curve: 7(1) = (7 +2r²)î + (21² – 4)ĵ + (7 –…

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Q: Find the length of one arch of the cycloid 1 8(t- sin(t)), y = 8(1 – cos(t)).

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Q: 2 Green's theorem, evaluate ((+) dx + 3xy dy where C is the circle * +y = 4.

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Q: Find the area of the surface generated when the given arc is revolved about the y-axis ㄷ y = 4 - x²…

A: We have to find area of the surface .

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: Use Green's theorem to evaluate $ 2y3 dx – x3y² dy, where C is the positively oriented circle of…

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Q: Write the equation of the curve l_r(t) which traces the circle of radius (0,0,1) on the plane z=1.…

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Q: ) Let C be the arc from the point (3,0) to the point (0,3) on the circle x² + y² = 9. The value of…

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Q: Express the circle specified by x 2 cos(0) y = 1+ 2 sin(0) in Cartesian coordinates and find the…

A: Note: Equation of a circle with radius r and center at (h, k) is given by (x-h)^2 + (y-k)^2 = r^2

Q: Express the circle specified by x = 2 cos(0) y = 1 + 2 sin(0) in Cartesian coordinates and find the…

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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: By using Cauchy's integral formulas evaluate sin Tz2 + cos TZ² dz (z + 1)(z – 2) C and e2z dz (z –…

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Q: O Find the length of arc of the curve y = from (0,0) to (4, ). %3D

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Q: 10. (~Ch16 Review #17) Use Green's Theorem to evaluate | x?y dx – xy? dy where Cis ,2 the circle x2…

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Q: Express the circle specified by x = 2 cos(0) y = 1 + 2 sin(0) in Cartesian coordinates and find the…

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Q: Verify Green's Theorem in the plane for F = -3y³i+ (3x3 + cos y)j where C is the circle x2 + y? =…

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Q: (a) Evaluate sin 2z dz, 3z -71 where C is an anticlockwise unit circle centred at the origin. 4.

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Q: sin z 0 dz, where C is the positively oriented circle of radius 1, |z| = 1. %3D

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Q: Q3- Use Green theorem to evaluate [7y-ex]dx+ [15x-sin(y³ + 8y)] dy Where (c) is a positively…

A: This is a application of integral problem

Q: Find the length of one arch of the cycloid x = 9(t – sin(t), y = 9(1 – cos(t)).

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Q: he length of the curve y = sin(3X) from x = 0 to I= is given by

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Q: Find the arc length of y = 7r +1 over the interval [0, 2]

A: y = 7r + 1 arc length=∫02 1+(dy/dr)2dr dy/dr = 7' arc length=∫02 1+(7)2dr arc length=∫02 50dr arc…

Q: Find the arc length over the domain 0<t<5 7t 1 10) x= V-7t – 2, y=- + | 4 2

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Q: Which of the following is an arc-length parametrization of the circle in the plane z=10 with radius…

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Q: Find an arc length parametrization of the line y = 8x + 7. r;(s) II

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Q: By revolving the semicircle x = r cos t, y = r sin t. (0 ≤ t ≤ π) about the x-axis, show that the…

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Q: Express the circle specified by x = 2 cos(0) y =1+2 sin(0) in Cartesian coordinates and find the…

A: Solution

Q: Consider the parametric curve: X = = 7et y = 3t5 Which integral gives the arc length of the curve…

A: I am going to solve the problem by using some simple calculus to get the required result of the…

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.