Let f(x, y) = ln(2x² + 8y² – 3). The level curve f (x, y) = 9 is: %3D An ellipse O A circle A parabola None of these

Q: Determine the exact area between the x-axis and the curve defined parametrically by 2 x(t) = - +² t²…

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Q: At what points does the curve r(t) = t i + (6t - t2)k intersect the paraboloid z = x2 + y2? (Enter…

A: Given: Equation of curve:                                    r(t)=t i +(6t-t2) k Equation of…

Q: The position of a particle in the xy-plane at time t is r(t) = (t+2) i+ (t- + 3) j Find an equation…

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Q: The diagram shows a sketch of the curve with equation y = -r +7x 12 1 3. -2- 8 CMA Q3 Find the…

A: For the solution follow the next step.

Q: If f'(x)= cosec* x-1 then what is the arclength of the curve y = f (x) on TC %3D V3 B) 3 A) 1 C) 2…

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Q: Evaluate SF dR given that F (x, y) = 2yî + (xy – 2x) %3D and C is the curve in the third quadrant…

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Q: 6. Find the positively oriented simple closed curve C for which th value of the line integral I 4 –…

A: Consider the given integral. I=∫Cy3−ydx−2x3dy Apply the green theorem. ∫CPdx−Qdy=∬D∂Q∂x−∂P∂ydA…

Q: 4. Let C be the unit circle in the xy-plane, and S be the boundary of the square with side 1,…

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Q: dy In the xy-plane, the point (0, 2) is on the curve C. If 4t for the curve, which of the following…

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Q: Let L be a tangent line to the hyperbola xy=4 at x=9 Find the area of the triangle bounded by L and…

A: Compute the tangent line as follows.

Q: Q3 Find dyldx on the graph y secx + y = cos2x at point (1, 2).

A: We have to find the derivative of above function at (1,2) Please see the explanation below

Q: Find . F dR such that F(x, y) = (2y, xy – 2x) And I is the curve in the 3rd quadrant from the point…

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Q: J (2x+y) dx + (x+2y) dy L is circle of center (7,8) and radius 9

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Q: Find the area of the surface generated when the given curve is revolved about the given axis.…

A: Given   Area of surface area revolved about x axis formula is

Q: The Normal to the curve 2x2 – y? = 1 at the point (1,1) is: %3D 3 y =-*+ y =x+ y = 2x + 1 %3D P. y =…

A: To find The equation of normal to the curve 2x^2-y^2=1 at the point (1,1)

Q: If f'(x)=Vcosec* x-1 then what is the arclength of the curve y= f (x) on TC TC A) 1 B) 3 C) 2 D) V3…

A: Given that, f'x=csc4x-1

Q: If f'(x) = /cosec*x-1 then what is the arclength of the curve y= f (x) on со ? 3 A) 1 В) 3 C) 2 D)…

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Q: a)Find the exact length L of the curve 3y = (4x – 3), 1 0. Answer: b) Evaluate xe*dx Answer: .3 c)…

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Q: Find the moment of inertia, with respect to the line x=1, of the area enclosed by the loop of the…

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Q: find ta Limits ofx i integual tahan 2*+34 dn dy as a) ネー15 5 h a) None of these

A: In the question we have to find the limits of x in the given function.

Q: tiny, no bug lands on a piece of graph paper and walks along a path so that t minutes after it…

A: Increasing function

Q: y = e*, 0<z<1.

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Q: Let f(x, y) = In(2x² + 8y² – 3). The level curve f(x,y) = 9 is: A circle A parabola None of these O…

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Q: Consider the tangent line to the curve r(t) = t2i −(1/(t + 1))j +(4 − t2)k at the point P(4, 1, 0).…

A: Given the tangent line to the curve r→t=t2i^−1t+1j^+4−t2k^  at the point P4,1,0. We have to…

Q: Let P(cosh t, sinh t) be a point on the unit hyperbola x2 − y2 = 1 in the first quadrant . Show that…

A: Redraw the given figure as shown below.

Q: The graph y = arctan x is transformed to y = a arctan(b (x – c)) + d by a vertical compression of…

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Q: Let f (x,y) = AS The level curve of f(x,y) = 10 is: %3D x2 +5y2" %3D O A circle An ellipse O A…

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Q: The graph y = arctan x is transformed to y = a arctan(b (x – c)) +d by a horizontal compression of i…

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Q: a) Find the length of the given curve: x = (y³/3) + 1/(4y) from y = 1 to y = 3 %3D b) Find the area…

A: a) given curve isx=y33+14y from  y=1,  y=3 find the length the curve  b) f(y)=4-y2, x=0 find the…

Q: Given the curve y = x´ and the lines y = 0 and y = 2 – x wwww Find the ordinate of the centroid Y חו…

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Q: Consider the curve x²y + y²x = 6 A) Find in terms of x and y dx

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Q: Suppose we have f (x, y) = x + 1' What is the level curve for f (x, y) = 5 ? (a) a parabola (b) a…

A: Let's find.

Q: Find two values of d such that the line 2x + y = d is tangent to the ellipse 4x2 + y? = 36. Find the…

A: follow next step

Q: Two particles move along the spatial curves r1(t) = (t, t2, t3) and r2(t) = (1 + 2t, 1 + 6t, 1 +…

A: In the question it is given that- r1(t)=t,t2,t3  and r2(t)=1+2t,1+6t,1+14t represents the spatial…

Q: Find the extreme value of the function f(x, y) = xyon the ellipse x2 + 2y2 = 1.

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Q: Let r(t) = -6t2 î+ (t² +3) ĵ + 7 k for t > 1. Reparameterize in terms of the arc-length parameter.

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Q: Given f(x, y) = x² - 4y, the parabola x² - 4y - 2 = 0 is the level curve of fat 02 0-2 01 00 None of…

A: Ggg

Q: Find the equation w=T(x,y,z) of the tangent hyperplane at p. F(x,y,z)=3x²-2y²+xz², p=(1,2,-1)

A: General equation for the tangent to the plane is expressed as:…

Q: The tangent and the normal to the parabola y²=20x at the point T(5,10) meet the x-axis at the points…

A: Details solution is given below:

Q: Find the orthogonal trajectories of the given family of curves. 1. y = c (sec x + tan x )

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Q: B. Find an equation of the tangent plane to the surface at the given point. 6. f(x,y) = x² – 2xy +…

A: The given surface is: f(x,y)=x2-2xy+y2Given point is:  P=(1, 2, 1)

Q: Find the area of the surface generated when the given curve is revolved about the given axis 1 = (e…

A: Given y=112e6x+e-6x  ,for -2≤x≤2  Area of surface of curve when revolved about x-axis…

Q: Find thề maximum height above the ry-plane of a point on r(t) = (e*, sin t, t( 14 – t)) The maximum…

A: NOTE: Refresh your page if you can't see any equations. . here we need to find the height above…

Q: Find two values of d such that the line 2x + y = d is tangent to the ellipse 4x2 + y2 = 81. Find the…

A: Given,           The line 2x + y = d is tangent to the ellipse 4x2 + y2 = 81.We know that,y = mx + c…

Q: Let f(x, y) = x? + 4y. Find the extreme values of f on the ellipse x? + 2y? = 1.

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Q: Evaluate | 9+x* dx+xy*dy along the positively oriented curve C where the C is the boundary of the…

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Q: c) By using Green's theorem evaluate $ (4xy – y°) dx + (x³ + 3xy) dy where C is a closed curve…

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Q: Use implicit differentiation to find all points on the graph of y* + y² = r(x – 4) at which the…

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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: