Surface Area of a Revolution

Hello all,

I need to find the surface area of the revolution of
http://latex.codecogs.com/gif.latex?\large%20y=xsin(x)
from -pi to pi about the x-axis. I've done this so far:

http://latex.codecogs.com/gif.latex?...sin(x)+xcos(x)
http://latex.codecogs.com/gif.latex?...2cos^{2}(x)}dx

I've tried a lot to simplify this and I've failed. Any ideas? Have I done something wrong?

Quote:

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Originally Posted by Ivanator27

Hello all,

I need to find the surface area of the revolution of
http://latex.codecogs.com/gif.latex?\large%20y=xsin(x)
from -pi to pi about the x-axis. I've done this so far:

http://latex.codecogs.com/gif.latex?...sin(x)+xcos(x)
http://latex.codecogs.com/gif.latex?...2cos^{2}(x)}dx

I've tried a lot to simplify this and I've failed. Any ideas? Have I done something wrong?

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surface area of rotation ...





I don't think this integral can be simplified using elementary methods ... may have to do the calculation w/ a machine

wolfram says ...

definite integral of x*sin(x)*sqrt(1+(sin(x)+x*cos(x))^2) from -pi to pi - Wolfram|Alpha

Hmmmm... is there not some elementary basis for finding the surface area of a revolution? Like the Riemann Sum for area under a curve. Perhaps I could use that?

Is there some reason not to use technology? If you insist, go with a Riemann sum ... however, you'll most probably have to use technology to calculate that, also.

I would like to know the principles involved with solving this problem, I don't mind using a computer or calculator if I understand what it's doing.

skeeter is suggesting a numerical integration. What you mention, the "Riemann sum", where you approximate the "area under the curve" as a rectangle (the function is approximated by piecewise constant), is one possibility. Another is the "trapezoid" method where you approximate the area by trapezoids (the function is approximated by piecewise linear functions). A third is "Simpson's rule" where the function is approximated by piecewise quadratic functions. You should be able to find all of those in any Calculus textbook.

http://latex.codecogs.com/gif.latex?...2cos^{2}(x)}dx

I'm familiar with Riemann sums and trapezium rule, so can I just use a Riemann sum to evaluate the expression being integrated? i.e. can I just do a Riemann sum using http://latex.codecogs.com/gif.latex?...x^2cos^{2}(x)} and the bounds pi and -pi? If i do so, could I post my working here so someone could check it?

Quote:

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Originally Posted by Ivanator27

http://latex.codecogs.com/gif.latex?...2cos^{2}(x)}dx

I'm familiar with Riemann sums and trapezium rule, so can I just use a Riemann sum to evaluate the expression being integrated? i.e. can I just do a Riemann sum using http://latex.codecogs.com/gif.latex?...x^2cos^{2}(x)} and the bounds pi and -pi? If i do so, could I post my working here so someone could check it?

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note that your integral represents the arc length of the curve, not the surface area of revolution as was stated in your original post.

Oops, is this better?

http://latex.codecogs.com/gif.latex?...2cos^{2}(x)}dx

Quote:

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Originally Posted by Ivanator27

Oops, is this better?

http://latex.codecogs.com/gif.latex?...2cos^{2}(x)}dx

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see post #2