Question

Vector calculus questions (attachment):

Answer 1

So for #16:

I parametrize x=t and y=t^2 so that we have vector r= <t, t^2>. Plugging in to get F(r(t))*r'(t) dt, I eventually end up with \[\int\limits_{-2}^{2}te ^{t^2}-2 + 2tsin(t^2) dt\]

And to be clear, when I wrote F(r(t))*r'(t) dt, I'm referring to a dot product. The integration's fairly straightforward, so I just want a second opinion of whether or not my steps up 'til that point were correct.

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For #17, I get z=g(x,y) = (1-x^2)^(0.5). I plug that in for z and multiply by ds.

ds is sqrt (dg/dx + dg/dy + 1).

Am I on the right track here? Sorry, I'm having a hard time with typing the math.

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#18

I parametrize x=cost, y = sint. vector r'(t) = <-sint, cost>. From there, I have no idea how to go on.

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#19

Haven't gotten to it yet.

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So there's no need for anybody to go through everything with me, but if somebody could nudge me in the right direction in any one of these problems (or ideally, tell me that I did #16 correctly), I'd appreciate it very much. I don't know why, but the more I try to do these types of problems, the more they escape me =(. Thanks a lot in advance!