Line Integrals I

Q. * The value of the line integral
$\int (2x{y}^{2}dx+2x^{2}ydy+dz)$ along a path joining the origin (0, 0, 0) and the point (1, 1, 1)

1. 0
2. 2
3. 4
4. 6

Q. Given a vector field $\overrightarrow{F}=y^{2}x{\hat{a}}_x-yz{\hat{a}}_y-x^2{\hat{a}}_z$, the line integral $\int \overrightarrow{F}.\overrightarrow{dl}$ evaluated along a segment on the x-axis from x= 1 to x= 2 is

1. -2.33
2. 0
3. 2.33
4. 7

Q. * The line integral $\int \overrightarrow{V}.d\overrightarrow{r}$ of the vector $\overrightarrow{V}=2xyz\hat{i}+x^{2}z\hat{j}+x^{2}y\hat{k}$ from the origin to the point (1, 1, 1)

1. is 1
2. is zero
3. is -1
4. can not be determined without specifing the path

Q. The line integral of the vector field F = 5xz$\hat{i}+(3x^2+2y)\hat{j}+x^{2}z\hat{k}$ along a path from (0, 0, 0) to (1, 1, 1) parametrized by $(t,t^2,t)$ is _______

1. 4.17

Q. F(x, y) = $(x^2+xy$) ${\hat{a}}_x$ + $(y^2+xy$) ${\hat{a}}_y$. It's line integral over the straight line from (x, y) = (0, 2) to (2, 0) evaluates to

1. -8
2. 4
3. 8
4. 0

Q. * The line integral of function F=yzi, in the counter clockwise direction, along the circle $x^2+y^2=1$ at z = 1 is

1. -2$\pi$
2. -$\pi$
3. $\pi$
4. 2$\pi$

Q. The line integral of the vector function $\overrightarrow{F}=2x\hat{i}+x^2\hat{j}$ along the x - axis from x = 1 to x = 2 is

1. 2.33
2. 3
3. 5.33

Q. The line integral ${\int }_{P_1}^{P_2}(ydx+xdy)$ from $P_1(x_1,y_1$) to $P_2(x_2,y_2$) along the semi-circle $P_1{P}_2$ shown in the figure is

1. $x_2{y}_2-x_1{y}_1$
2. $(y_2^2-y_1^2)+(x_2^2-x_1^2)$
3. $(x_2-x_1$)$(y_2-y_1$)
4. $(y_2-y_1{)}^2+(x_2-x_1{)}^2$

Q. Consider points P and Q in the x-y plane, with P =(1, 0) and Q = (0, 1).The line integral 2${\int }_P^Q$(xdx+ydy) along the semicircle with the line segment PQ as its diameter

1. is -1
2. is 0
3. is 1
4. depends on the direction (clockwise or anitclockwise) of the semicircle)

Q.

Estimate the value of $\cos 2\mathrm{\pi }?$

Q. * A path AB in the form of one quarter of a circle of unit radius is shown in the figure. Integration of $(x+y{)}^2$ on the path AB traversed in a counter-clocwise sense is

1. $\frac{\pi }{2}-1$
2. $\frac{\pi }{2}+1$
3. $\frac{\pi }{2}$
4. 1

Q. $\text{Given a vector field}\overrightarrow{F}=yx{\hat{a}}_x-y^{2}z{\hat{a}}_y-2x^2{\hat{a}}_z,\text{the line integral}\int \overrightarrow{F}.\overrightarrow{dl}\text{evaluated along a segment}\text{on the x-axis from}x=1\text{to}x=3\text{is}$

1. 0
2. 1.66
3. 3.33
4. -3.33