Evaluate the line integral, where C is the given curve; rds; where C:x =t°,y = t, 0

Q: You are buying a car. Would you prefer a higher rate for a lower term or a lower rate for a longel…

A: Given that You are buying a car. Would you prefer a higher rate for a lower term or a lower rate for…

Q: Find the tangent/s as directed:to the parabola (x + 2y)^2 + 2x – y – 3 = 0, parallel to the line x –…

A: Given equation of line is  x-4y=24y=x-2 y=14x-12 So comparing with the slope equation formula,…

Q: 5) Sketch the curve X = e', Y = e?t + 1

A: Given, The curve  X = et  and  Y = e2t+1 We have to sketch the curve.

Q: 1) Find the length of the curve? X = y 3 + -| 1 4y from y=1 to y ==

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Q: 3. Find the points where tangent lines are vertical or horizontal: x³ + y³ = 6xy

A: Given:- x3+y3 = 6xy To find :- Points where tangent horizontal and vertical

Q: Which of the following lines is perpendicular to the tangent line of y= х-3 at the point (2, –2) ? y…

A: We have to solve

Q: 2. For what values of the numbers a and b is the curve f(x) = eaz²+bx tangent to the line y = e at…

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Q: The tangent line to the curve r(t) =(t°,t“, t³ ) at the point (1, 1, 1) intersects the æy-plane at a…

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Q: Find the equation of the tangent plane to z = e + x + x* + 10 at the point (4, 0, 79). Submit answer

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Q: 6. Find the tangent line and normal line to curve y = x3 - 3x2 - 2 at point (1, -4)?

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Q: d two coordinate points at which the tangent line to the curve given by x² – 2x + 4y² = 15 is…

A: if the tangent is vertical then dydx will be ∞x2-2x+4y2=15differentiate 2x-2+8ydydx=04ydydx=1-xsince…

Q: 11. Find the equation of the tangent line at (2, 1) given In y = 4y² – x³ +4

A: 11. The equation of the curve is given by lny = 4y2-x3+4 To evaluate : The equation of the tangent…

Q: Evaluate the length of the curve x = 6 – Vy – 2 , 3< y<6

A: To find the length of following curve  x = 6 - sqrt( y - 6) 3 ≤ y ≤ 6

Q: 1) Find the length of the curve? X = y 3 3 + 1 4y from y = 1 to y = 3

A: we have given curve

Q: Let C be the curve consisting of line segments from (0, 0) to (1, 1) to (0, 1) and back to (0, 0).…

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Q: Find an equation of the tangent plane to: 3x +y = z at P(-1,1,-2) %3D

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Q: Find the length of the curve y = V- from z =1 to r = e. 3

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Q: Let F = (4x³ + yz, xz, xy). Let C be any smooth curve from (1, 1, 1) to (2, 3, 0) Calculate cF dr.

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Q: C is the part of the circle of radius 3 in the first quadrant from (3,0) to (0,3) and F = (1, -y),…

A: Given, C is part of circle of radius 3 in the first quadrant from (3,0) to (0,3) and F=[1,-y] The…

Q: At the point (a, b) the curve T =t-4t, y=t2 +2t, t eR has a vertical tangent line. What is the sum a…

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Q: The tangent to the curve f(x) = V9 – x² at the point where x = B makes an angle of 30° measured with…

A: Using the Slope of tangent =dy/dx. and slope of tangent = tan(@)

Q: 3. For the curve x – xy² = 8e²4, do the following | a) Verify that the point (2,0) is on this curve.

A: Given, curve x3-xy2=8e2y

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: