6. Let D be the region bounded by the parabola y = x² and the line y = 1. Let I be the boundary of D. Compute f,ī dz, where z is the conjugate of z, by (a) using the definition of the line integral, and (b) using Green's Theorem.

Q: Calculate the line integral f -y²dx + xdy where ŕ is the path y² = 2x − x² such that x>1 and y > 0.

A:

Q: 1. Use Green's theorem in order to compute the line integral $ (3cos + 6y?) dx + (sin(5") + 16x³) dy…

A: Here the given integral is                               ∮C3cosx+6y2dx + sin5y+16x3dy

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

A:

Q: 3. Let Cibe the circle radius 2 centered at the origin, oriented counterclockwise, let C₂ be the…

A:

Q: n exact form. Use symbolic notation and fractions where needed.)

A:

Q: the plane for (x³ - x²y)dx + xy²dy], where (C) is the boundary of the region enclosed by the circler

A: The solution is given by

Q: Use Green's Theorem to evaluate the following integral Let² dx + (5x + 9) dy Where C is the triangle…

A:

Q: Compute the line integral: y² dx + x² dy two ways. C

A: We have an line integralTo evaluate the line integral by two methods (a) By  Green theorem(b) By,…

Q: Show that the function given by f(x,y)= x² + 3xy is differentiable at every poin using the…

A: Introduction: Distinctiveness (differentiability) implies continuity. If a function is…

Q: Describe and draw at least 3 level surfaces for the function f(r, y, z) = x²+ y? - z?

A:

Q: Given the line integral | (2x2° + 6y) da + (6x – 2yz) dy + (3a² z² – y²) dz. Let C be the path that…

A: Note that the following integral reduces like this :

Q: Let the curve C be the line segment from (0, 0) to (3, 1). Let F = ⟨2x-y, 4y-x⟩ Calculate the…

A:

Q: Evaluate the circulation of G = ayi+zj+ 7yk around a square of side 9, centered at the origin, lying…

A:

Q: 2.Proof that any tangent plane for the surface F(- point = 0 passses through a fixed

A: Given below clear explanation

Q: Find the area between the curve of y ਆਗੂਆਂ 2 4 = zæ + 2 and the x-axis on the interval [-2, 1].

A: Given that , Now we want to find the area between the curve and the axis  on the interval .

Q: Use the cross product to find the area of the portion of the plane defined by z = x + y which lies…

A:

Q: Evaluate the line integral where C is the curve made up of the straight line segments from the…

A: We will find parametric equations of line segment and then find line integral

Q: Calculate the line integral f (2x²y + 10x + 6y − 6) dx + (−6xy + 7) dy, - where C' is the rectangle…

A:

Q: (a) Let f(2) = 24 +5z3 Evaluate the integral f(2)dz, where the contour C is the circle k| = 2.

A:

Q: Use Green's Formula to evaluate the line integral along the positively oriented closed curve C:

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: