PYQs Based Test - 09 : Line, Surface, and Volume Integrals

Given $\vec{F}=\left(x^2-2y\right)\vec{i}-4yz\vec{J}+4x{z}^2\vec{k},$ the value of the line integral _C^ F.dl along the straight line C from (0, 0, 0) to (1, 1, 1) is.

Determine the volume of the solid of revolution formed when the curve y = 2 is rotated 360° about the x-axis between the limits x = 0 to x = 3.

The surface integral $\int \underset{s}{\int }\frac{1}{\pi }\left(9xi-3yj\right)$.nds over the sphere given by x² + y² + z² = 9 is

The volume under the surface $\mathrm{z}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{x}+\mathrm{y}$ and above the triangle in the x-y plane defined by $\{0\le \mathrm{y}\le \mathrm{x}\mathrm{and}0\le \mathrm{x}\le 12$ } is _________.

The integral ${\oint }_c\left(ydx-xdy\right)$ is evaluated along the circle $x^2+y^2=\frac{1}{4}$ traversed in counter clockwise direction. The integral is equal to