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Integrals, Green's Theorem, Positively Oriented Curve, Ellipse, Vector Equation and Surfaces

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Evaluate the line integral by two methods: (a) directly and (b) using Green's Theorem.
∫c xdx + ydy. C consists of the line segments from (0,1) to (0,0)...and the parabola y = 1 -x^2....

Use Green's theorem to evaluate the line intgral along the positively oriented curve.
∫c sin y dx + x cos y dy C is the ellipse x^2 + xy + y^2 = 1

Identify the surface wth the given vector equation.
r(x,&#952;) = (x, xcos&#952;, xsin&#952;)

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Integrals, Green's Theorem, Positively Oriented Curve, Ellipse, Vector Equation and Surfaces are investigated. The solution is detailed and well presented.

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Solution.

Method 1. Directly calculation.

So,

Method 2. use ...

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• BSc , Wuhan Univ. China
• MA, Shandong Univ.
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• "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
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