Find the work done by the force field F(x,y)=x2i+yexj on a particle that moves along the parabola x=y2+1 from (1, 0) to (2, 1)

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Jack Maxson · Accepted Answer

Step 1&amp;nbsp;By definition&amp;nbsp;The line integral along a path is the work done along that path.That is Work done&amp;nbsp;=∑CF⋅dr&amp;nbsp;Step 2&amp;nbsp;Note that&amp;nbsp;dr=&amp;nbsp;dx&amp;nbsp;i+&amp;nbsp;dy&amp;nbsp;j&amp;nbsp;Given that&amp;nbsp;F(x,y)=x2i+yexj&amp;nbsp;Therefore, Work done&amp;nbsp;=∑C(x2i+yexj)⋅(&amp;nbsp;dx&amp;nbsp;i+&amp;nbsp;dy&amp;nbsp;j)&amp;nbsp;Work done&amp;nbsp;∑Cx2&amp;nbsp;dx&amp;nbsp;+yex&amp;nbsp;dy&amp;nbsp;&amp;nbsp;Step 3&amp;nbsp;Given that&amp;nbsp;C:x=y2+1&amp;nbsp;Therefore,&amp;nbsp;y=x−1&amp;nbsp;&amp;nbsp;and&amp;nbsp;&amp;nbsp;&amp;nbsp;dy&amp;nbsp;=&amp;nbsp;dx&amp;nbsp;2x−1&amp;nbsp;In order to completely express an integral in terms of x, swap out the values of y and dy.Work done&amp;nbsp;=∑Cx2&amp;nbsp;dx&amp;nbsp;+x−1ex&amp;nbsp;dx&amp;nbsp;2x−1&amp;nbsp;Work done&amp;nbsp;=∑Cx2+12ex&amp;nbsp;dx&amp;nbsp;&amp;nbsp;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&amp;nbsp;=∑12x2+12ex&amp;nbsp;dx&amp;nbsp;&amp;nbsp;=[x33+12ex]12&amp;nbsp;=[233+12e2]−[133+12e1]&amp;nbsp;=83+12e2−13−e2&amp;nbsp;=73+e2−e2

Jeremy Merritt · Accepted Answer

Call the parabola P, parameterized by r(y)=⟨y2+1,y⟩ with 0≤y≤1⟩. Then the work done by f(x,y) along P is  ∫Pf(x,y)⋅dr=∑y=0y=1f(x(y),y)⋅dr(y)dydy  =∫01⟨(y2+1)2,yey2+1⟩⋅⟨2y,1⟩dy  =∫01(2y5+4y3+2y+yey2+1dy=73−e2+e22

karton · Accepted Answer

Step-by-step explanation: By definition: Work done along the path is the line integral along that path denoted as: Work done =∫CFdr Note: dr=dxi+dyj Given that: F(x,y)=x2i+yexj F(x,y) dot product with dr=x2dx+yexdy Work done =∫C(x2dx+yexdy) Eq1 Given that CL y=x−1 dy=dx2x−1 Replace the value of y and dy in Eq 1 Workdone =∫C(x2+ex2)dx Limits of x are 1 to 2 respectively Workdone =∫12(x2+ex2)dx =(x33+ex2)12 Evaluate limits to obtain Work Done =73+e2−e2

Find the work done by the force field F(x,y)=x^{2i}+ye^{xj} on

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