PYQs Based Test - 10 : Application of Theorems

The value of the integral

${\oint }_s\vec{r}.\vec{n}ds$

over the closed surface S bounding a volume V, where $\vec{r}=xi+yj+zk$ is the position vector and n̂ is normal to the surface S, is

The line integral of function $F=yz\hat{i}$, in the counter clock wise direction, along the circle $x^2+y^2=1$ at $z=1$ is

As shown in the figure, 𝐶 is the arc from the point (3, 0) to the point (0, 3) on the circle 𝑥² + 𝑦² = 9. The value of the integral $\underset{c}{\int }\left(y^2+2yx\right)dx+\left(2xy+x^2\right)dy$ is _____ (up to 2 decimal places).

If C represents a line segment between (0, 0, 0) and (1, 1, 1) in the Cartesian coordinate system, the value (expressed as an integer) of the line integral ${\int }_C\left[(y+z)dx+(x+z)dy+(x+y)dz\right]$ is________.

The vector function F(r) = -x î + yĵ is defined over a circular are C shown in the figure.



The line integral of ∫${}_C$ F(r) ⋅ dr is