Question

Evaluate the line integral ${\mathbf{\int }}_{\mathbf{c}}{\mathbf{y}}^{\mathbf{2}}\mathbf{dx}\mathbf{+}\mathbf{2}\mathbf{xdy}\mathbf{+}\mathbf{dx}$ where Cconnects $\left(0,0,0\right)\mathbf{}\mathbf{with}\mathbf{}\left(1,1,1,\right)$

(a) Along straight lines from $\left(0,0,0,\right)\mathbf{}\mathbf{to}\mathbf{}\left(1,0,0\right)\mathbf{}\mathbf{to}\mathbf{}\left(1,0,1\right)\mathbf{}\mathbf{to}\mathbf{}\left(1,1,1,\right)\mathbf{;}$

(b) on the circle ${\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}\mathbf{-}\mathbf{2}\mathbf{y}\mathbf{=}\mathbf{0}\mathbf{}\mathbf{to}\mathbf{}\left(1,1,0\right)$and then on a vertical line to$\left(1,1,1\right)$.

Short answer

The solution to this problem is mentioned below.

a) $\frac{3}{8}$,

b)$\frac{8}{3}$ .

Step-by-step solution

Given Information.

The given information is ${\int }_c{y}^{2}dx+2xdy+dz$.${\int }_c{y}^{2}dx+2xdy+dz$.

Definition of Line integer.

A line integral is integral in which the function to be integrated is determined along a curve in thecoordinate system.In mathematics and physics, scalar field or scalar-valued function referred to a scalarvalueto everypointin aspace– possiblyphysical space. The scalar may either be a (dimensionless)mathematical numberor aphysical quantity.

Find the solution of (a) part.

a) In order to get from the origin to the point (1,1,1) take an infinite number of routes.

One single route from the origin to the point (1,0,0) on a straight line. In this segment the variable x changes continuously from 0 to 1, but the other two variables remain 0. This means that the line integral is zero.

y=z

=dy

=dz

=0

The second displacement is from the point (1,0,0) to the point (1,0,1) in this segment z changes continuously from 0 to 1 , but the variable x is fixed to the value 1 , which means

dx=0,

y=dy

=0

This means that the last contribution is not zero.

$$\begin{gathered} {\int }_{c}dz={\int }_0^{1}dz \\ =1 \end{gathered}$$

The last segment is from (1,0,1) to (1,1,1) in this segment the variables x and z are fixed at the value 1 , which gives,

dz=dx

=0

The variable y changes continuously from 0 to 1.

$$\begin{gathered} {\int }_{c}2xdy={\int }_0^{1}2.1dz \\ =2 \end{gathered}$$

The total line integral is ${\int }_c{y}^{2}dx+2xdy+dz=3.$

Find the solution of (b) part.

b)Take the curve C to go from the origin to the point (1,1,0) as a circle.

$x^2+y^2-2y=0$ …….(1)

The equation (1) can be written in different form.

$x^2+{\left(y-1\right)}^2-1=0$

This is a circle with radius 1 and center at y=1 .

Write the coordinates in trigonometric form.

$$\begin{gathered} x=\cos \theta , \\ y-1=\sin \theta . \end{gathered}$$

Simplify the coordinates.

$$\begin{gathered} y=1+\sin \theta \\ {y}^2=1+2\sin \theta +{\sin }^2\theta \\ dx=-\sin \theta d\theta \\ dy=\cos \theta d\theta \end{gathered}$$

Here, $\theta$changes from $\frac{3\pi }{2}$ to $2\pi$.

Evaluate the integral.

role="math" localid="1659172605940" $$\begin{gathered} {\int }_{\frac{3\pi }{2}}^{2\pi }d\theta \left(1+2\sin \theta +{\sin }^2\theta \right)\left(-\sin \theta \right)+2\cos \theta \left(\cos \theta \right)={\int }_{\frac{3\pi }{2}}^{2\pi }\left(-\sin \theta -{\sin }^3\theta +2\cos \left(2\theta \right)\right)d\theta \\ =\cos {\overline{)\theta }}_{\frac{3\pi }{2}}^{2\pi }-{\overline{)\frac{1}{12}\overline{)\left(\cos \left(3\theta \right)-9\cos \left(x\right)\right)}}}_{\frac{3\pi }{2}}^{2\pi }{\overline{)+\frac{2}{2}\sin \left(2\theta \right)}}_{\frac{3\pi }{2}}^{2\pi } \\ =1-\frac{1}{12}\left(1-9\right) \\ =\frac{5}{3} \end{gathered}$$

The last segment is a straight line from (1,1,0) to (1,1,1) where z changes continuously from 0 to 1 , and x and y are fixed at a value 1 , which gives,

$$\begin{gathered} dx=dy \\ =0 \end{gathered}$$

Therefore, the last segment is equal to 1 .

$$\begin{gathered} {\int }_c{y}^{2}dx+2xdy+dz=\frac{5}{3}+1 \\ =\frac{8}{3} \end{gathered}$$

Hence,the solution to this problem is mentioned below.

a) 3 ,

b) $\frac{8}{3}$.