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Problem 41 Hard Difficulty

Find the work done by the force field $$ \textbf{F}(x, y, z) = \langle x - y^2, y - z^2, z - x^2 \rangle $$ on a particle that moves along the line segment from $ (0, 0, 1) $ to $ (2, 1, 0) $.

Answer

Work done $=\frac{7}{3}$

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Video Transcript

So you know, first we are paramo triceps. Um, extra beauty to t. Why should be t c should be one minus. T t from 0 to 1 is an extended parametric ization of the segment. So if of our tea, simply replace x by two t. Why buy tea? So why is she z square? Is this zebra Z replaced by one minus t and the minus X square to T minus T square, T minus one. So this one should be negative. T square plus three T minus one and nearly 40 square minus She plus What? Let me check it again. Now The expansion should be correct. So we take the dot product as are we have the first period of what is our priority. We take the derivative component wise. So what is the dot product? 40 minus two T square, minus T squared. Plus three T minus one plus 40 square plus T minus one. This search. Yeah. Mm. So let's see how we get a negative two t square Negative T square prosper 40 square. So T square 40. Prosperity is 17. Lost his 18. So minus one and minus one. So minus two so we just have to. That's an open a new page we integrate from 0 to 1, which is a range of T T squared plus 18 minus two. And I mean we can just write down the answer 1/3, because the T cube over three plumbing one plus. Similarly, this is 40 square plugging. One should be four to t plus one to t and plugging one should be should be, too. UH, so two plus 1/3 7/3.