LINE INTEGRAL AND GREEN'S THEOREM – NAT - LEVEL–2

Evaluate the following integral along segment of straight line joining the given points

$\int xdx+ydy+(x+y-1)dz$ from (1, 1, 1) to (2, 3, 4)

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Let $l$ be the length of the portion of the curve $x = x(y)$ between the lines $y = 1$ and $y = 2$ where $\dfrac{dx}{dy}=\dfrac{\sqrt{1+y^2+y^4}}{y}, \ x(1)=0$

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The value of the $\underset{C}\int \vec{F}\cdot d\vec{r}$ for $\vec{F}=x^2\hat{i}+xy\hat{j}$ and C is the curve $y^2 = x$ joining (0, 0) to (1, 1) is : (correct upto three decimal places)

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The arc length of the curve $x = e^t \sin t, \ y = e^t \cos t$ from $t=0$ to $t=\dfrac{\pi}{2}$ is (upto three decimal places) :

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Let $\vec{F}(x,y)=(2ye^{xy}+2xe^{x^2-y^2})\hat{i}+(2xe^{xy}-2ye^{x^2-y^2})\hat{j}$ where C is the curve $y = x^2$ from (0, 0) to (1, 1) such that $\int _{C}\vec{F}\cdot d\vec r$ is equal to $k(e - 1)$ than $k$ equals

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Evaluate $\int \vec{F}\cdot d\vec{r}$ where $\vec{F}=(x-2y)\hat{i} +(y-2x)\hat{j}$ along the closed curve C in xy plane given by $x = \cos t, y = 2 \sin t$ from $t=0$ to $t=2\pi$ .

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Find the value of $\int (x^2+y^2)dy$ taken the counter clockwise since along the quadrilateral with vertices (0, 0), (2, 0), (4, 4), (0, 4). (Upto three decimal places)