Given the line integral | (2x2° + 6y) da + (6x – 2yz) dy + (3a² z² – y²) dz. Let C be the path that consists of the line segment from from A(1; 0; 5) to B(1; 0; 3), followed by the line segment from B(1; 0; 3) to C(2; 1; 1), and the line segment from C(2; 1; 1) to D(2; 2;1).

Q: af 11. Suppose + = 0, prove that the line integral | F. fe -dx + ду dr = -dy is inde- pendent of…

A: Here F = fy i+fxj

Q: Evaluate the line integral 2'y dx +a/y² +1 dy for C : r(t) = (VE, t?),0 < t < 1 and then repeat with…

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Q: 2. Evaluate the given line integral, | (-y,x – y) · dR, where C is the line segment from (1,0) to…

A: To find the line integral we use the equation of line segment in parametric form using given points.

Q: Use Green's Theorem to evaluate the line integral. cx2 - y?) dx + 5y? dy C: x² + y² = 9 (x² .

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Q: Evaluate the triple integral. 2xy dV, where E is bounded by the parabolic cylinders y = x² and x =…

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Q: The integral ∫C[(3x2y+y2)dx+(x3+2xy)dy] is independent of the path. Evaluate the integral where C is…

A: Given: R=ti+(t2+t-2)j   for 0&lt;=t&lt;=2

Q: Find the line integral ye-xz ds Here, C is the path r(t) = [t,3t,-6t], 0sts In(8) ·

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Q: If F= 2y I- 3J +x²k and s the surface parabolic cylinder y= 8x in bounded by the planes OF the 2.…

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Q: Evaluate the integral (2x – y) dx + (x + 3y) dy along the path C. C: x-axis from (0, 0) to (0, -3)…

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Q: Verify Green's Theorem by evaluating both integrals y2 dx + x² dy = dA ду for the given path. C:…

A: In this question we can verify the Green's Theorem . i.e, ∫C M dx +Ndy = ∫∫R ∂N∂x-∂M∂ydA .

Q: 2. Evaluate the given line integral, (-y,x – y) · dR, where C is the line segment from (1, 0) to…

A: Introduction: The formula of line integral of the function f(x,y) on the line segment C(t) is given…

Q: evaluate both the line integral and the double integral. 16. P(x, y) = 2x – x³y°, Q(x, y) = x³y°, C…

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Q: Use Green's Theorem to evaluate the line integral. (a) | e2a+y dx +e¬Y dy, where C'is the triangle…

A: In the question we have to evaluate integral. NOTE - As per Bartleby's policy we can not solve…

Q: Evaluate the line integral fay dx + x dy where C is the parameterized path a = t2, y = t³, 2 <t < 5.…

A: We have to find line integral in this question.

Q: Evaluate the line integral [x(1 с where C is given by the line r(t) = (1 + 3t, 2t), 0≤ t ≤ 1. x(1 +…

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Q: Suppose F(x, y) = (6x + 5y)i + (5x + 4y)j. Evaluate the line integral for each of the given paths,…

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Q: 1. Let C be a path along the upper half of a circle from the point P(2,0) to the point Q(-2,0).…

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Q: Compute the line integral -xdy + ydx 2(x-y)2 +4xy where L is the circle (1-x)2 + y2 = 3 with a…

A: We can write the given line integral as ∫L y2x-y2+4xydx+-x2x-y2+4xydy, where L is the circle…

Q: 3. Calculate the line integral S F - d7, where F = (x²y³, x³y?) and C is the path given by C 7(t) =…

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Q: Evaluate the triple integral (x + 2y) dV where E is bounded by the parabolic cylinder y = 7x² and…

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Q: Verify Green's Theorem by evaluating both integrals v² dx + x² dy = / / dA ax for the given path. C:…

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Q: valuate the line integral Scy da +a dy where C is the parameterized path a = t2, y = t°, 1<t<4. Scy…

A: We have to find the given integral.

Q: Use Green's Theorem to evaluate the line integral (y – x)dx + (2x – y)dy for the given path. C: x =…

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Q: Set up an integral for the length of y = vx + 2, 2 2 <x<7 using ds = 1+ ) dy. A. ds = f, /1+ 4y²dy…

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Q: Evaluate the line integral S, y ds where C:x = t²,y= 2t, 0 <t < 3

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Q: The line integral 2xe" dæ + a²e" dy + (1 – 2²)e¬²² dz is not independent of path.

A: The line integral ∫CF·dr is independent of path if the curl of F is zero, that is, ∇×F=0. Here, The…

Q: Use Green's Theorem to evaluate the line integral Jo (y - x)dx + (2x - y)dy for the given path. C: x…

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Q: Evaluate the line integral (e*y+ 2x) dx + (e* + 1) dy, where C is a smooth curve from the point (0,…

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Q: Show that the integral is independent of the path, and use the Fundamental Theorem of Line Integrals…

A: Line integral will be path independent if vector field whose integral we need  is conservative…

Q: Evaluate the line integral Scy dx + x dy where C is the parameterized path x = t°, y = t³, 1 < t< 3.…

A: follow next step

Q: Evaluate the line integral cy dx + x dy where C is the parameterized path x = y = t° , 2<t< 6. Scydx…

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Q: Evaluate the line integral H. dř; Ħ = e" ì + [æe" + In(z)] ĵ +2² k, and C is the path defined by 7…

A: Given H→=eyi^+xey+lnzj^+2yzk^ We know that dr→=dxi^+dyj^+dzk^…

Q: 2. Evaluate the given line integral, (-y,x – y) · dR, where C is the line segment from (1,0) to…

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Q: 1. Evaluate the line integral of E = xax - y ay along the segment P₁ to P2 of the circular path as…

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Q: Evaluate the line integral T, (“+y) dx + 4 ry dy where L is the path, -> r) = (3 t, 2) with 0 <t <1.…

A: We evaluate given integral

Q: Evaluate the line integral di the path over C from to Q(2,4) P(O,) where F(xy) 2xy°2+3y5 13D to ネソーメ…

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Q: Evaluate the line integral Sa y dx +z dy, where C is the line segment from (4) (-1, 1, 2) to (2,3,…

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Q: Evaluate the line integral |(2? + y) dx – x dy where C is the line segment from (2,5) to (5,9) in…

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Q: Evaluate the line integral ∮C x dy−y dx where C is a circular, closed path defined by x^2+y^2=16

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Q: Evaluate the line integral. y dx + z dy + x dz, r(t) = (2 + t-1, t3, t2) for 0 <t< 3

A: Here, x=2+t-1 y=t3 z=t2

Q: 4. Evaluate the line integral along the path C given by x = 2t, y = 10t, where 0 ≤ t ≤ 1. I (a) √(x…

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Q: Evaluate the triple integral -p (x + 6y) dV where E is bounded by the parabolic cylinder y = 8x² and…

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Q: Suppose F(x,y)=(4x+6y)i +(6x+4y)j. Evaluate the line integral for each of the given paths, which are…

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Q: Use the Fundamental Theorem of Line Integrals to find the exact value of the path-independent line…

A: First of all hare we check for the path independent line integral then solve for moving over the…

Q: Use Green's Theorem to evaluate the line integral (у — х) dx + (2х — у) dy for the given path. C:…

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Q: Evaluate the line integral 9x²yzds with respect to s along the 2 curve C: x=t, y = t², z = t³(0 ≤ t…

A: Given: ∫C9x2yzds x=t y=t2 z=23t3 0≤t≤1 To find: Value of the given integral.

Q: Verify Green's Theorem by evaluating both integrals aM | y? dx + x² dy = dA ax for the given path.…

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Riemann Sum

Riemann Sums is a special type of approximation of the area under a curve by dividing it into multiple simple shapes like rectangles or trapezoids and is used in integrals when finite sums are involved. Figuring out the area of a curve is complex hence this method makes it simple. Usually, we take the help of different integration methods for this purpose. This is one of the major parts of integral calculus. 

Riemann Integral

Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.