Q6. Determine whether F (x, y) is conservat integral F dr, given that; F(x,y,z) = (1+ xy)e*y i + x²e*Y j %3D and C: 7(t) = cost i – 2sint j, 0

Q: Compute the line integral of the scalar function f(x, y) = /1+9xy over the curve y = x³ for 0 < x <…

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Q: Evaluate the line integral f(2x+4y)ds for C: r(t)= ⟨3t2, 4t2+3⟩, 0 =&lt; t =&lt; 2

A: Now, we can compute this integral. We'll need to multiply  (22t2+12)(10t),  integrate with respect…

Q: B- Evaluate the integral 1-x (3-x-y dz dy dx /1-x²

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Q: Let C be the parabola y = x² for 0 < x < 2. Let F(x, y) = ye"i + xe*"j.Compute the line integral

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Q: Use Green's Theorem to evaluate the line integral § 2y² dx + 3x² dy, where C is the boundary of the…

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Q: Evaluate the line integral Jc F · dr, where F(x, y, z) = xi – yj + 3zk and C is given by the vector…

A: Given F(x,y,z)=xi-yj+3zk and C is the curve given by  rt=sint,cost,t. Find ∫CF·dr.

Q: Evaluate (x + y)ds where C is the straight-line segment x=t, y = (4 – t), z = 0 from (0,4,0) to…

A: We have to find the given line integral.

Q: Use the Fundamental Theorem of Line Integrals to evaluate F. dR where x 1 (−2x + 1)²³ y +1 and C is…

A: Given: F→x,y=lny+1-1-2x+12, xy+1+1y2+1 C is the portion of the parabola y=-x2+1 from 1,0 to 0,1.…

Q: ¸R = {(x, y) = R² | 4 ≤ x² + y² ≤ 9} Let's look at the level area and vector field F(x, y) = -5yi +…

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Q: x=3 Qs. A: Find * x²dx by using Riemann integral.

A: Hi! Thank you for the question, As per the honor code, we are allowed to answer one question at a…

Q: x+ 1 y ds C along the curve C given by r (t) = (2 sint, 2 cost), 0<t< m

A: Given integral is ∫Cx+12yds. Given vector is, rt=2sint,2cost,0≤t≤πxt=2sint,yt=2cost

Q: Let F(x, y) = (2x + e – y cos? x, xel + sin y). Let C1 be the path from (0,0) to (7,0) along the…

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Q: Find the derivative of the function f(z, u) = =*y – 4y° at the point Po(2, 1) in the direction of =…

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Q: B) Evaluate the line integral of the function P(x,y) dx + Q(x,y) dy. over the curve C where…

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Q: {(x, y) E R²; \x| + \y – a| < 1} andI= (y – 2x)2 y2 + 1 -dxdy. (i) Sketch the graph of the domain D…

A: We are given D=x, y∈ℝ2; x+y-x≤1  and  I=∬Dy-2x2y2+1 dxdy. For (i), Here, we need to sketch the graph…

Q: Consider the following. C: counterclockwise around the circle  x2 + y2 = 49  from  (7, 0) to (−7, 0)…

A: Given,   C: Counterclockwise around the circle            x2+y2=49       from (7, 0), and (-7, 0).…

Q: Compute the line integral of the scalar function f(x, y) = √√/1 + 9xy over the curve y = x³ for 0 ≤…

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Q: F(x, y, z) = (2xe*) i+ (2y + 3y?22)j+ (x²e* + 2y³z) k is conservative. Find a functionf such that F…

A: To find the function f  And then evaluate the line integral along the given curve.

Q: Evaluate F· dr along the curve C. F(x, y) = x² i+xy j C:r(t) = 2 costi+2 sintj (0 < t < n)

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Q: Compute the line integral of the scalar function f(x, y) = /1+ 9ry over the curve y = x° for 0 <¤ <…

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Q: [f.dr, where W and f = (x² + y 1V

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Q: Let F(x,y,z) = (y,x²,zy) where C:R(t) = (cost,sin t , 3t) and t e [0,2]. Evaluate %3D %3D •dR

A: We have to find value of given Integral.

Q: (a) Verify Green's theorem for the line integral f(x−y)dx + (x+y)dy, where C C is the unit square…

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Q: Consider the following functions. F(x, y, z) xi - zj + yk r(t) = 4ti + 9tj - t²k 16² F. dr, where C…

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Q: Find the line integral (4х — 5у) ds, where C is the semicircle y = 9 – x² traversed from (3,0) to…

A: Graph the semi-circle. Take x=3cost and y=3sint Now, x'=-3sint and y'=3cost The angle covered by…

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.