Use Green's theorem to evaluate the line integral I of the one-form W = + æ sin? (y)) dæ + (x² cos(y) sin(y) + xy+sin" (y)) dy along the closed curve in R? formed by going from the origin to the point (T,0) along the arc of the curve y = 8 sin(x), and then back to the origin along the a-axis.

Q: 6. Consider the parametric curve x(t) = 1 + cost+cos 2t, y(t) = sint + sin 2t = sin t(1+2 cos t)…

A: Given that                Then            At  , we need to find the value of .

Q: Let ŕ ). ) - (0**) k (4 Sin 2+) + (4 cos 2 t, be the Position oF Perticle space in at time. Finc the…

A: The velocity is dr/dt and the acceleration is d2r/dt2

Q: Let y tan 1)2x then dy/dx at x-1 is

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Q: Find y SX a) y = 3 sin (x² + 5x - 3) y' = c) y = sin" (6x) y'= b) xE5X+2 d) y = In (2x) Y₁ = Y y' =

A:

Q: Which of the following gives the length of the curve y x2 + 2x from x = 0 to x = 5 ? (x2 + 2x) dæ I…

A: Length of the given curve will be

Q: Find the general solution to the linear first-order differential equation: -+ dx 3x x²+1= 1 6x x²+1

A: The given differential equation,.The objective is to find the general solution to the differential…

Q: find dylde O x+ tanxy =D0

A: In the question it is asked to calculate derivative of x+tan(xy)=0 with respect to x.…

Q: Find the unit tangent vector to the curve defined by r(t) = (4 cos(t), 4 sin(t), 2 sin²(t)) π at t =…

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Q: consists of y = 4-x and straight line from (-2,0) to (2,0). Sketch the curve C. 64 ans: 3

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Q: Let C be the curve given by X- Sin/tr) and y=t, from ĭst 55. sin/tT and y=t, from ist E5. Find the…

A: According to Fundamental Theorem of Line Integral, the definite integral of a function is evaluated…

Q: Calculate the line integral [F•dr, where F = 3x°yi + x'j in two cases for C1 and C2. These two…

A: The given vector is F=3x2yi+x3j Here we use Green's theorem. So the Green's theorem is…

Q: 0.8 0.6 0.4 0.2 of -0.2 -0.4 -0.6 0.5 1.5 2 5 A direction field for y'= -(2xy2 + y³) 2.5

A: Given differential equation is: y'=-2xy2+y3. We have to sketch the direction field for the given…

Q: Write parametric equations for a(t) at (1,1) a(t) =

A:

Q: Find the slope of the tangent to the curvey=cos²x-sin'x at x- O a) 1 O b) 1/1/2 Oc) -2 d) 2

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Q: 6a) Let A be the parabolic arc defined by the equation y = 8x – x´ over the domain interval [0, 8].…

A: (a) Consider the given function. y=8x-x2 Now, draw the above function by using the calculator.

Q: sin(2t), y = – cos(2t), z = 4t at P(0, 1, 27) has equation 87 + 2 sin(1). The normal plane to the…

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Q: Find the slope of the tangent line to the parametric curve 2te x(t) y(t) at the point P(0, 1). 2t +e…

A: xt=2t etyt=2t+etAt  P 0,1,  x=0,   y=1  ∴x=0⇒2t et=0             ⇒t=0   dxdt=ddt2t et= 2 t…

Q: For the curve r-y +2xy-y+1=0: a) Find the derivative dy/dx. b) Where do the horizontal tangent lines…

A: follow the next step

Q: Solve the initial value problem x'(t)=Ax(t) for t20, with x(0)=(5,4). Classify the nature of the…

A:

Q: The parametric equation of the tangent line to the curve C defined by the following parametric…

A: given: the parametric equation of the curve C is given by: where . we have to find the parametric…

Q: 1- Find dB/dx for the follow ing functi'on a)- xy = Cot (xy) b- y Sim(hs =-メy %3D c)- y?cos () 2x+2y…

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Q: (x² + y?)3/2' + (x² + y?)3/2J OVer the curve F(t) = (e' cos(t))i+ (e' sin(t))j from the point (1,0)…

A:

Q: Prove that all normal to the curve x =acid+at sin t, Y=a sin t-atcost.Are at a distance a from the…

A: We need to prove that normal to the given curve are at a distance of a from origin.

Q: 2- Find the equation of the tangents to curve given parametrically by Y= sin(x²) + 1 at (x,y)= (…

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Q: Does f(z)=z/(sin z)^2 have a pole of order 1 or 2 at z=0?

A: f(z)=zsin2zFor the poles first we look at the zeros of the denominatorat z=0 We know sin(z) has a…

Q: Find the length of the curve r(t): (4 cos(t) + 4t sin(t))i + (4 sin(t) - 4t cos(t))j + (3-√3²) 2 k…

A: