(1,0) (1,0) | (x2 – y²) dx – 2x y dy + i | 2x y dx + (x² – y²) dy (0,1) (0,1) Calculate the above integral along the given curve and the result, where a, b, c, d negative integers, b Find the numbers so that. i x? + y? = 1 1

Q: f f(x,y) = 3x² – 9y², then the total differential of f at z = 1 and y = -2 is :

A:

Q: z = f(x, y) = +x*e +In[(2x + y)*I 2xy Determine the total differential (dg), and total derivative…

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Q: X Find the length of the curve y = cos 2t dt from x= 0 to x = 4 Set up the integral to find the…

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Q: Using a Differential as an Approximation, f(x, y) = 2x − 3y find f (2, 1) and       f (2.1, 1.05)…

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Q: Let y 2x2. Find the change in y, Ay when x 1 and Ax 0.3 Find the differential dy when z = 1 and dx…

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Q: Q3/Find the value of integral along the curve C joining the points (1, 1) and (2, 3) | (12z2 – 4iz)…

A: We have to find the value of integralalong the curve C joining thepoints (1, 1) and (2, 3).

Q: Assume that F= (e^x−y)i+ (2y−x)j, C1 is curve from (0,1) to (2,5) along the graph of y=x^2+ 1, and…

A: Given , F=ex-yi^+ (2y-x)j^ C1 is a curve from (0,1) to (2,5) along the parabola y=x2+1 C2 is a line…

Q: Q.1) (a) Evalute (x² – y²)dx + 2xydy, along the circular arc C by x = cost, y: sint (0 <t <). (b)…

A: answer

Q: Double integral of x^3 cos(x^2y) dxdy ? for x=1/y, x=1/2 & 1<y<2

A:

Q: Show that the form under the integral sign is exact in the plane and •(2,2) evaluate the integral /…

A: Given the line integral is ∫-1,-12,2e-x2-y2xdx+ydy. The line integral∫CF·dr=∫CF1dx+F2dy.

Q: 2. Triple Integrals For any continuous function f and constant a > 0, show that cx 1 II Fl0) dz dy…

A: Just integrate one by one.

Q: px+y e- dt = xy %3| dy dx |(x,y)=(0,0) y satisfies the given integral. Find the derivative of y at…

A:

Q: cln(2) cln(5) 2x+y dy dx Evaluate the iterated integral:

A:

Q: Find the work integral of F = ⟨y^2, x⟩ along the line segment C starting from P(0, 1) and ending at…

A:

Q: integral(integral e^(x/y) dy from x to 1)dx from 0 to 1

A: Given:

Q: Using a Differential as an Approximation f(x, y) = x cos y, find f (2, 1) and f (2.1, 1.05) and…

A: the given function is z=f(x,y)=xcosy

Q: show that the differential forms in the integrals are exact. Then evaluate the integrals.

A: To Determine: show that the differential forms in the integrals are exact. Then evaluate the…

Q: Q:4) Evaluate the line integral of the function p(x.y)=y-x,over the straight line Joining the two…

A: Function p(x, y ) = y - x   points ( 4, 0 ) and ( 0, 4 )  To Evaluate: Integral  of the given…

Q: 19. Determine the line integral of the differential form (x + 3y²)dx − (−6xy+z)dy from points…

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Q: Evaluate the iterated integral. 7 2 z z [ [²] ² 1 -1 J-y Y (-72+3 ln (7)) x dx dz dy

A:

Q: e the Fundamental Theorem of Line Integrals to calculate F. dr exactly. F- 4 sin(4x + y)7 + sin(4x +…

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Q: Calculate the iterated integral integral_0^2 integral_0^4 (6 x ^2y - 3 x) dy dx

A: by solving given Integral

Q: (1,0) (1,0) 1 = | (x² – y²) dx – 2 x y dy + i - 2 x y dx + (x² – y²) dy (0,1) (0,1) Calculate the…

A:

Q: 4. the value of the given integrals | z - i | = 2,(z € C) around the circle compute for positive…

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Q: 2- 3- Evaluate the integrals in Exercises [² dx L₁3 3 dx 3x - 2 dy 1- 2. S sec y tan y 2 + secy

A: We can answer all the three questions as below.

Q: 3) S. (3x? – 3yz + 2xz)dx + (3y? – 3xz + z²)dy + (3z2 - 3xy + x2 + 2yz)dz Calculate the curvilinear…

A:

Q: Calculate the iterated integral integral_1^4 integral_0^2 (6 x y^2 - 2 x) dy dx

A: the function given to us is  ∫14∫02(6xy2-2x)dy dx

Q: Q2 Use Green's theorem to evaluate the line integral F•ds where F = 2xyi +(x-y')j and C is the path…

A: In vector calculus, Green's theorem relates a line integral around a simple closed curve C to a…

Q: Evaluate the line integral for the following problems over the given regions. Use the above theorems…

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Q: (1,0) (1.0) |(r? - y?) dx + 2xy dy + i | -2xy dx + (x² - y') dy (0,1) (0,1) Calculate the above…

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Q: dx a) Evaluate the integral / z In(4x) using the substitution u = In(4x). dx x In(4x) +C b) Evaluate…

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Q: 2 - 4 f(2)dz = / f(x)dx + | f(=)da - f(t)dt, а 9 9 what are the bounds of integration for the first…

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Q: y=x² from (0,0) to (1,1). | Compute the line integral of f(x)=2x over the curve 2009

A:

Q: a) Evaluate the Integrals: 2x2 + 3 x(x + 1)2 b) Find a horizontal line y = k that divides the area…

A: (a) part Consider the given integral ∫2x2+3xx+12dx Now,…

Q: х — 0, у- 1, x = y7 about the line y = 1 can be computed using the method of disks or washers via an…

A: Given:- x = 0, y = 1, x=y7 To find :- Volume of solid about the line y = 1

Q: Set up an integral that gives the length of the curve y2 + y = 2x from the point (1,1) to (3,2)

A:

Q: Q.1 Find the double integral le cos x² dA Where R between : x=-y/2, x=1 and y=-2 , y=0. Q.2 Sfr…

A: Please see the explanation below

Q: Find the differential of the function f(x,y)=−3x2−5xy−9y2f(x,y)=-3x2-5xy-9y2 at the point…

A:

Q: Q#4. Is Green's theorem applicable for all types of line integrals? Explain clearly. Use Green's…

A: We will use greens theorem

Q: A bird of mass m flying along a path given by r(t) = (2t, t^2, t) subject to the force of gravity…

A: r(t)=(2t, t2, t) subject to the force of gravity pulling it towards the origin.  The force is F(x,…

Q: (c) Determine the arbitrary constant of integration for the integral: y = [(4t -5-dt given that y =…

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Q: Evaluate the line integral / -ydx + xdy along the curve C : y? = 6x from (6,6) to (0,0). NOTE: Enter…

A: We have to find the line integral ∫C-ydx+xdy along the curve C: y2=6x from (6,6) to (0,0).

Q: x € [0,2] n Q1 x € [0,2|\QS Find upper and lower Reimann integral for T(x) =

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Q: 2 V2y-y2 Evaluate the integral x²+ y²dzdydx by using cylindirical coordinates.

A:

Q: integral(integral e^(x/y) dy from x to 1)dx from 0 to 1

A: To compute the double integral.

Q: Let I = L S x+1 хdydx. %3D -1J-V1-x By reversing the order of integration of I, we obtain: Two…

A:

Q: Using a Differential as an Approximation, f(x, y) = 16 − x2 − y2 find f (2, 1) and f (2.1, 1.05) and…

A: Given function is: f(x, y) = 16 − x2 − y2 f (2, 1) = 16 - 22 − 12 f (2, 1) = 16 - 4 - 1 f (2, 1) =…

Q: d) Use line integral method to evaluate F(x, y)• dř ,where R 1 F(x, y) = (3x y°i + 4xy² j) and M, N,…

A: The graph of curves y=x2, y=x is as follows The yellow colored region is region R

Q: If f(x, y) the total differential of f at (0, ) i 16e Cos x + 7 cot y, then the total differential…

A: Total differential

Q: dz (x2²–x³) 1/3dx Using the substitution *=-, dx= - the integral reduces to x3 (22– 1) 1/3 -dz A) -…

A:

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: