Evaluate F.dr where F(x, y) = xy i + e*j and C is the line segment from (2, 0) to (4, 0). а) 0 b) 2 с) - 2 d) 4 e) – 4

Q: 4) For which one of these linear maps T:R → R? is det T > 0? a) T(1, y) = (y, -r) b) T(1, y) = (x,…

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Q: Calculate fo(7x + 6y) i + (4x + 7y) j) dr where C is the circular path with center (a, b) and radius…

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Q: z dv, where E is enclosed by the paraboloid z = x2 + y² and the plane z = 4. Evaluate

A: see 2nd step

Q: 2 Consider the line L: y -1 +t|0, te R, and the plane T : x – 4y – z = 6. (a) Show that L is…

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Q: Evaluate the line integral xyz ds where C is the line segment joining the points (1, – 1, 1) to (1,…

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Q: Sſcos( dxdy y-x y+x D

A: Given and u=y-x,  v=y+x where D is the trapezoid with vertices (1, 0), (2, 0), (0, 2) and (0, 1).

Q: Evaluate ſſ, Vy² – x² dA where boundary of R is the parallelogram bounded by the four lines

A: In this question, we evaluate the double integral by using the change of variable. The region is…

Q: Evaluate //. dV, where E x2 + y² + lines between the spheres x + y + z? and x + y? + z2 = 25 in the…

A: Consider the provided question, Evaluate ∫∫∫E1x2+y2+z2dVwhere, E lines between the sphere x2+y2+z2=1…

Q: If C is the line segment from (1, 3) to, (0, 0), find the value of the line integral: 3y i+ 3x j) ·…

A: Given If C is the line segment from (1, 3) to, (0, 0),

Q: Show that the value of ∮C xy2 dx + (x2y + 2x) dy around any square depends only on the area of the…

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Q: If C is the line segment from (1, 3) to, (0, 0), find the value of the line integral: So(3y² i +3xj)…

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

Q: Evaluate the integrals

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Q: Evaluate the line integral So 7y dr + 3x dy where C is the straight line path from (2, la 7uda + 3z…

A: Introduction: A line integral is an integral that determines the function to be integrated along a…

Q: WX || ZY and Zs Z and Y are supplementary WXYZ is a parallelogram Given: Prove: W Y

A: It is given that,  WX ¯|| ZY¯  and ∠Z and ∠Y are supplementary To prove :- WXYZ is a parallelogram

Q: Let d: Rx R*→[0, ∞) such that d(XF)=√(x, y)² show that (R.d) metric space.

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Q: Calculate f((5x + 2y)i + (4x + 7y) 3) · dr where C' is the circular path with center (a, b) and…

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Q: Find a parametrization c(t) of the line y = 5 − 2x such thatc(0) = (2, 1).

A: This question comes under the category of calculus and from the parametrization of the line here we…

Q: Verify Green's theorem for Q7 [x(x? + y²)dx – y(3x² – y²)]dy Where C is the boundary of the triangle…

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Q: Suppose that Vf(x, y, z) = 2xyze*i+ ze**j+ ye*´k. If f(0, 0, 0) = 5, find f(1, 1, 8). Hint: As a…

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Q: soùd lines For Portions dash ùnes of borndry included in thee bmain and not include Suot1 For d)…

A: We are asked to find the domain of the function.

Q: 38. Calculate y = x, x = 0, and x = 1. || e* dA, where D is bounded by the lines y =x +1,

A: We have to find ∫∫D ex dA, where D is bounded by the lines y=x+1, y=x, x=0 and x=1

Q: Jo xy ds where C is the line segment from (2,0) to (5, 4). Evaluate the line integral

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Q: a) Evaluate f, (2x – z) dS, where S is part of the plane 3x + 2y + z = 1 that lies above the…

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Q: f (lnx + y) dx − a² dy, where & is the rectangle with vertices (1, 1), 1. Use Green's Theorem to…

A: Let I =∮(ln x+y)dx-x2 dy, where the region is C which is a rectangle with vertices…

Q: 4. If M is a centroid of triangle WOR and WM = 16, what is WX? %3D W Y M R

A: centroid:- i). It is the intersection point of all the medians of a triangle. ii). It divides the…

Q: If C is the line segment from (6,3) to, (0,0), find the value of the line integral: Se(by? i+ 3z) -…

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Q: Use the Green's theorem to calulate So (3x²y – xy? – 3.x²y²) dx + (x³ – 2.x³y+ x²y) dy, where C is…

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Q: Let C be the positively oriented square with vertices (0, 0), (2,0). (2, 2). (0, 2). Use Green's…

A: Green's Theorem:  For a positively oriented curve C∫CP dx+Q dy=∫∫D∂Q∂x-∂P∂y dAfor the region D…

Q: || x dA, R is bounded by y = x, 2x + y = 6 and the y-axis. R

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Q: In APQR, V is the centroid. If QU=54 find QV. S T V R U

A: Centroid is the point of intersection of the medians.we know that centroid divides the each median…

Q: 1 Evaluate dV, where E lines between the spheres a? + y? + z² = 25 and %3D x² + y? + z2 49 in the…

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Q: Use Stoke 's Theorem to evaluate [ V. dr, where v= y²i + xyj + xzk, and c is the bounding curve of…

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Q: Integrate f(x,y,z) = (x + y +z)/ (x² +y² +z? over the path r(t) =t i+tj+3t k, 0<a<t<b.

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Q: Compute where h(s) = rs? -r. Assume r is a constant. %3D ds

A: Since you have asked multiple questions in a single request, we would be answering only the first…

Q: Show that the function u = p(ry) + JFÿ¢(y/x) provides the quation „Pu = 0. dy

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Q: 2. Calculate (y+ z)dx+(x+z)dy + (x+ y)dz , where C consists of line segments from (0,0,0) to (3,2,4)…

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Q: Consider z = Ax + By + D be the tangent plane to f(x, y) = 2x² + 5y³ + 9xy + 9x + 3y at (6,9).What…

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Q: If C is the line segment from (1, 5) to, (0,0), find the value of the line integral: So(4y? i + 4x…

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Q: Let f(x, y, z) = eª + zy² and let C be a straight line segment connecting the points (4,2,1) and…

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Q: Dsig Laplace transfoem, るe the d撃.e ざくE)+4y +3yCt) =et, yiven that yo)=1, の=. y o)=1, し. Ans yCt) =…

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Q: 5. Show that the equation of the tangent plane to f(r, y, z) = c at (ro, Yo: 20) is given by the…

A: This is the question from vectors.

Q: Let W = Span{u} where u -2. Let y = |1, find the distance from y to W L3]

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Q: If C is the line segment from (8, 1) to, (0,0), find the value of the line integral: Se(8y² i+x3) ·…

A: First we have to write the equation of the line segment C and then convert the given line integral…

Q: Evaluate dV, where E lines between the spheres a+ y + z = 9 and E 2 + y2 + z? x2 + y? + z2 = 16 in…

A: The answer is in step 2

Q: Show that F(x, y, z) = (y² – e* sin z)i + (2xy + z)j + (y – e* cos z)k is conservative and hence…

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Q: Note: Answer either Yes or No or Not-Applicable in the block provided. Consider the DE ry" – y' +y =…

A: We will find out the required values.

Q: a) Evaluate the line integral S. (2x - y) dx – (x + 3y) dy, where C is a straight line from (1,1) to…

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Q: Use t = y to parametrize the part of the line of intersection of the two planes y = 2x – 4 and z =…

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Q: Evaluate F dr where C is the line segment from (7,-,1) to (20, 1, and F (sin(yz) + 1)i + (zz cos(yz)…

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Arc Length

Arc length can be thought of as the distance you would travel if you walked along the path of a curve. Arc length is used in a wide range of real applications. We might be interested in knowing how far a rocket travels if it is launched along a parabolic path. Alternatively, if a curve on a map represents a road, we might want to know how far we need to drive to get to our destination. The distance between two points along a curve is known as arc length. 

Line Integral

A line integral is one of the important topics that are discussed in the calculus syllabus. When we have a function that we want to integrate, and we evaluate the function alongside a curve, we define it as a line integral. Evaluation of a function along a curve is very important in mathematics. Usually, by a line integral, we compute the area of the function along the curve. This integral is also known as curvilinear, curve, or path integral in short. If line integrals are to be calculated in the complex plane, then the term contour integral can be used as well.     

Triple Integral

Examples: