Find the center and radius of circle x“ + y° - 2x-4y+1=0. Prove thot (0

Q: Given line YZ tangent to OJ at point Y and mzWYZ = 119, what is the measure of arc WXY %3D Y W X.

A: Measurement of arc is the angle subtended by arc at the center of circle.

Q: Evaluate COS Z -dz , where C is the circle |z – 1| = 3. (Integrate counterclockwise around C.)

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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: (b) Find the circle of curvature at (0, 1)of the curve y = x3 + 2x² + x +1

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Q: 10. Evaluate , (2x – y) dA where D is bounded by the circle with center the origin and radius 2.

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

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Q: Q2 Calculate 8- z dz, z(4- z) where C is the circle of radius 7, centre 0, negatively oriented.

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

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Q: Q2 Calculate 8- z dz, z(4-z)* where C is the circle of radius 7, centre 0, negatively oriented.

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Q: If C'is the circle of radius 3 centered at the origin, starting at the point (3,0) and moving around…

A: Answer and explanation is given below..

Q: .If C is the circle (traversed once counterclockwise) of radius 2 with center at z = 0, %3D 7e?…

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Q: PR is tangent to circle Q at R and PS Is tangent to circle Q at S.Find mZP.

A: Given: PR is tangent to Q at R and PS is tangent to Q at S ∠P =  x° ∠Q =  4x° ∠R =  90° ∠S =   90°

Q: 1. Integrate P (3)*dz, C: clockwise unit circle centered at (0, 0).

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Q: Q2 Calculate 8-z dz, z(4-z) where C is the circle of radius 7, centre 0, negatively oriented.

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Q: Use Green's Theorem to evaluate the integral (3x² – y)dx + 3xdy where C is the circle a? + y? = 1…

A: As we know from the Green's theorem; ∫CMdx+Ndy=∫∫D∂N∂x-∂M∂ydA   Given; I=∫C3x2-ydx+3xdy So, M=3x2-y…

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: 3. Find the area of the surface generated by revolving the curve y = 2/x ,1Sx<2, about the x-axis(as…

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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: 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 curvature and radius of curvature of the plane curve at the given value of x. y = sin 2x, x…

A: Consider the provided equation, y=sin2x,   x=π4 Find the curvature and radius of curvature of 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: 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: Evaluate o y dx – x³ dy, where C includes the two circles of radius 2 and radius 1 centered at the…

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Q: B. Find the length of the circle of radius r defined parametrically by x = r(v1 – sin t) and y =…

A: We need to find the length of the circle which is parametrically defined as below.

Q: A circle of radius a in the xy plane, centred at the c is described by T = ¢ +acos (t) i +asin (t)…

A: In this question, we need to change the orientation of the path.

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

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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: Compute ž dz +y1 z dz, +72 and where +Y1 is the segment from 1 to -1, and +y2 is the arc on the unit…

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Q: Determine the osculating circle of the helix defined by y = x^3 − 3x + 1 at x = 1.

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Q: s) Evaluate the line itegral ( +2y + 2)dy – dz, where C is the arc of parabola z = 2+ y". from (2,0)…

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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: Integrate counterclockwise dz, C' the circle with z² + 8z + 15 center -3 and radius 2. dz = La z2 +…

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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: 5. Evaluate (y +tan x)dx+(x -2ry+y)dy, where C is the circler +y' =4 and is oriented in a positive…

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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: Use Green’s theorem to evaluate the line integral 2x2y dx + 6x3 dy, where C is the circle x2+y2 = 4,…

A: The answer is as follows:

Q: Q2 Calculate dz, - z) where C is the circle of radius 7, centre 0, negatively oriented.

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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: ) + () 2 If C is the part of the circle = 1 in the first quadrant, find the following line integral…

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Q: Q2 Calculate 8- Z -dz, (4 – 2) where C is the circle of radius 7, centre 0, negatively oriented.

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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: Find the radius R, of curvature to the hyperbola y = x + 3x² – x³ at the point x = 0 given that R =…

A: To find the radius R, of curvature to the given hyperbola at point x = 0.

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

A: Solution

Q: If C is the circle of radius 3 centered at the origin, starting at the point (3, 0) and moving…

A: -81π/2

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.