The value of the line integral \(I = \int \vec{A}\cdot d\vec{r}\), where \(\vec{r} = x\hat{i} + y\hat{j}\) and \(\vec{A} = (x+y)\hat{i} + (y-x)\hat{j}\), along the path \(y^{2} = x\) from the point \((1, 1)\) to the point \((4, 2)\) is:
11.33
Parametrise the path \(y^{2} = x\) as \(y = t\), \(x = t^{2}\) with \(t\) going from \(1\) to \(2\); then \(dx = 2t\,dt\) and \(dy = dt\).
The integrand becomes \((x+y)\,dx + (y-x)\,dy = (t^{2}+t)(2t)\,dt + (t-t^{2})\,dt = (2t^{3} + t^{2} + t)\,dt\).
Integrating from \(1\) to \(2\), \(I = \left[\dfrac{t^{4}}{2} + \dfrac{t^{3}}{3} + \dfrac{t^{2}}{2}\right]_{1}^{2} = \dfrac{34}{3} \approx 11.33\).
Hence, the value of the line integral is 11.33.
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.................... is a client server application that identifies each host on the internet with a unique user-friendly name.
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