lunnatican4

2022-01-06

Find the work done by the force field $F(x,y)={x}^{2i}+y{e}^{xj}$ on a particle that moves along the parabola $x={y}^{2}+1$ from (1, 0) to (2, 1)

Jack Maxson

Beginner2022-01-07Added 25 answers

Step 1

By definition

The line integral along a path is the work done along that path.

That is Work done $=\sum _{C}F\cdot dr$

Step 2

Note that $dr=dxi+dyj$

Given that $F(x,y)={x}^{2}i+y{e}^{x}j$

Therefore, Work done $=\sum _{C}({x}^{2}i+y{e}^{x}j)\cdot (dxi+dyj)$

Work done $\sum _{C}{x}^{2}dx+y{e}^{x}dy$

Step 3

Given that $C:x={y}^{2}+1$

Therefore, $y=\sqrt{x-1}\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}dy=\frac{dx}{2\sqrt{x-1}}$

In order to completely express an integral in terms of x, swap out the values of y and dy.

Work done $=\sum _{C}{x}^{2}dx+\sqrt{x-1}{e}^{x}\frac{dx}{2\sqrt{x-1}}$

Work done $=\sum _{C}{x}^{2}+\frac{1}{2}{e}^{x}dx$

Since x increases from 1 to 2 as we proceed from (1,0) to (2,1), we shall integrate from 1 to 2 with respect to x.

Work done $=\sum _{1}^{2}{x}^{2}+\frac{1}{2}{e}^{x}dx$

$={[\frac{{x}^{3}}{3}+\frac{1}{2}{e}^{x}]}_{1}^{2}$

$=[\frac{{2}^{3}}{3}+\frac{1}{2}{e}^{2}]-[\frac{{1}^{3}}{3}+\frac{1}{2}{e}^{1}]$

$=\frac{8}{3}+\frac{1}{2}{e}^{2}-\frac{1}{3}-\frac{e}{2}$

$=\frac{7}{3}+\frac{{e}^{2}-e}{2}$

Jeremy Merritt

Beginner2022-01-08Added 31 answers

Call the parabola P, parameterized by $r\left(y\right)=\u27e8{y}^{2}+1,y\u27e9$ with $0\le y\le 1\u27e9$ . Then the work done by f(x,y) along P is

${\int}_{P}f(x,y)\cdot dr=\sum _{y=0}^{y=1}f(x\left(y\right),y)\cdot \frac{dr\left(y\right)}{dy}dy$

$={\int}_{0}^{1}\u27e8{({y}^{2}+1)}^{2},y{e}^{{y}^{2}+1}\u27e9\cdot \u27e82y,1\u27e9dy$

$={\int}_{0}^{1}(2{y}^{5}+4{y}^{3}+2y+y{e}^{{y}^{2}+1}dy=\frac{7}{3}-\frac{e}{2}+\frac{{e}^{2}}{2}$

karton

Expert2022-01-11Added 613 answers

Step-by-step explanation:

By definition:

Work done along the path is the line integral along that path denoted as:

Work done

Note:

Given that:

F(x,y) dot product with

Work done

Given that CL

Replace the value of y and dy in Eq 1

Workdone

Limits of x are 1 to 2 respectively

Workdone

Evaluate limits to obtain

Work Done

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