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

The volume enclosed by the surface $f\left(x,y\right)=e^x$ over the triangle bounded by the line $x=y;x=0;y=1$ in the xy plane is ________.

Let aₓ and a${}_y$ be unit vectors along x and y directions, respectively. A vector function is given by

F = aₓ y - a${}_y$ x

The line integral of above function

${\int }_{C}F\cdot dl$

Along the curve C, which follows the parabola y = x² as shown below is ______ (rounded off to 2 decimal places)

A vector field D = 2ρ²a${}_{\rho }$ + z a${}_z$ exists inside a cylindrical region enclosed by the surfaces ρ = 1, z = 0 and z = 5. Let S be the surface bounding this cylindrical region. The surface integral of this field on $S\left(\int \int ◯{}_{S}D.ds\right)$ is _______.

The volume determined from ∫∫∫v 8 xyz dv for V = [2, 3] × [1, 2] × [ 0,1 ] will be (in integer) ________.

Consider the line integral $I=\underset{C}{\int }\left(x^2+i{y}^2\right)dz$, where z = x + iy. The line C is shown in figure below.



The value of I is