Calculate c fr dr for the given data if f is a force this...

Calculate ∫C F(r) • dr for the given data. If F is a force, this gives the work done by the force in the displacement along C. Show the details.F = [ex, cosh y, sinh z], C: r = [t, t2, t3] from (0, 0, 0) to (1/2, 1/4, 1/8). Sketch C.

Transcript

00:01 So, we are given that the function f is equal to e to the power x comma cosine hyperbolic y comma sine hyperbolic z and r is equal to t comma t square comma t cube.

00:23 So, now first of all we find out f of r dr.

00:27 So this can be represented as e to the power x dx plus cosine hyperbolic y into dy plus cosine hyperbolic z into dz.

00:40 Let's take the closed integral on both sides f of r dr.

00:45 So this should be equals to the closed integral of this whole term e to the power x dx plus cosine hyperbolic y dy plus the sine hyperbolic z dz.

00:56 Z so this can be separately done so now as we can see r is equals to what t comma t square comma t cube so plug into above we get x is equals to t y is equals to t square and z is equals to t cube so can i say dx is equals to dt dy is equals to 2t dt and dz is equals to 3t square dt so plug into to above our integral will become integration 0 to 2 sorry 0 to t and e to the power in the place of x we have to put t and here it will become d in the place of bx as dt in similar way it will become integral 0 to t cosine hyperbolic t at 2t dt plus sine hyperbolic a t integral 0 to t 3t square dt.

01:52 Now separately we have to find this integral.

01:55 So as per the question t is given to us t is equals to 1 half.

02:00 So apply this trouble our integral of f of r dr would become equals to integration 0 to 1 by 2 e to the power t dt plus 2t cosine sign hyperbolic t plus 3t square sign of hyperbolic t and dt here it will be so just cancel it we have taken it separately so now we can integrate it so this will become equals to the integration of c 0 to 1 by 2 e to the power t will dt will be as it is now what will be the integration of these two terms this will become a c 2 will be common a t sine hyperbolic t minus cosine of hyperbolic t how did we do that because see there were two of the first term is adjusted in the second one that is why we have written in this way plus of for the second term it will be a 3 a t square plus 2 cosine of hyperbolic t minus 2t sine of hyperbolic t and the limit will go from 0 to 1 over 2.

03:14 Finally, this will be goes to e to the power t integral the limit 0 to 1 by 2 plus 2 times of t sine hyperbolic t minus cosine of hyperbolic t limit will go from 0 to 1 by 2 plus of 3 t square plus 2 cosine of hyperbolic t minus 2t sine hyperbolic t limit will go from 0 to 1 by 2...

Question

Calculate ∫C F(r) • dr for the given data. If F is a force, this gives the work done by the force in the displacement along C. Show the details.F = [ex, cosh y, sinh z], C: r = [t, t2, t3] from (0, 0, 0) to (1/2, 1/4, 1/8). Sketch C.

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