lunnatican4

2022-01-06

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

Jack Maxson

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
Given that $F\left(x,y\right)={x}^{2}i+y{e}^{x}j$
Therefore, Work done
Work done
Step 3
Given that $C:x={y}^{2}+1$
Therefore,
In order to completely express an integral in terms of x, swap out the values of y and dy.
Work done
Work done
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
$={\left[\frac{{x}^{3}}{3}+\frac{1}{2}{e}^{x}\right]}_{1}^{2}$
$=\left[\frac{{2}^{3}}{3}+\frac{1}{2}{e}^{2}\right]-\left[\frac{{1}^{3}}{3}+\frac{1}{2}{e}^{1}\right]$
$=\frac{8}{3}+\frac{1}{2}{e}^{2}-\frac{1}{3}-\frac{e}{2}$
$=\frac{7}{3}+\frac{{e}^{2}-e}{2}$

Jeremy Merritt

Call the parabola P, parameterized by $r\left(y\right)=⟨{y}^{2}+1,y⟩$ with $0\le y\le 1⟩$. Then the work done by f(x,y) along P is
${\int }_{P}f\left(x,y\right)\cdot dr=\sum _{y=0}^{y=1}f\left(x\left(y\right),y\right)\cdot \frac{dr\left(y\right)}{dy}dy$
$={\int }_{0}^{1}⟨{\left({y}^{2}+1\right)}^{2},y{e}^{{y}^{2}+1}⟩\cdot ⟨2y,1⟩dy$
$={\int }_{0}^{1}\left(2{y}^{5}+4{y}^{3}+2y+y{e}^{{y}^{2}+1}dy=\frac{7}{3}-\frac{e}{2}+\frac{{e}^{2}}{2}$

karton

Step-by-step explanation:
By definition:
Work done along the path is the line integral along that path denoted as:
Work done $={\int }^{C}Fdr$
Note: $dr=dxi+dyj$
Given that: $F\left(x,y\right)={x}^{2}i+y{e}^{x}j$
F(x,y) dot product with $dr={x}^{2}dx+y{e}^{x}dy$
Work done $={\int }^{C}\left({x}^{2}dx+y{e}^{x}dy\right)$ Eq1
Given that CL $y=\sqrt{x-1}$
$dy=\frac{dx}{2\sqrt{x-1}}$
Replace the value of y and dy in Eq 1
Workdone $={\int }^{C}\left({x}^{2}+\frac{{e}^{x}}{2}\right)dx$
Limits of x are 1 to 2 respectively
Workdone $={\int }_{1}^{2}\left({x}^{2}+\frac{{e}^{x}}{2}\right)dx$
$=\left(\frac{{x}^{3}}{3}+\frac{{e}^{x}}{2}{\right)}_{1}^{2}$
Evaluate limits to obtain
Work Done $=\frac{7}{3}+\frac{{e}^{2}-e}{2}$

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