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Algebra/Integration the volume

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Expert: - 8/30/2009


Question

Question diagram
The diagram shows the curve y=(e^-.5x) (1+2x)^.5 and its maximum point M. The shaded region between the curve & the axes is denoted by R. Find by integration the volume of the solid obtained when R is rotated completely about the x-axis. Give your answer in terms of pi & e. I got this pi*INT of ((e^-x)(1+2x)dx) but I dont know how to continue the rest using integration by parts & using the limits by integrating twice. I tried many times already each time I tried got different answers. Can you please help me? Thanks a lot

Answer
Hello Warm,

Looks like at least your formula is correct. I'm guessing you know that the limits of integration are x=-0.5 to x=0. Right?

Something you can do right away is split your expression into two integrals.:

V=pi*INT of ((e^-x)(1+2x)dx) = pi*INT of ((e^-x)dx) + 2*pi*INT of (x*(e^-x)dx).

You can integrate the first one right away, cause the integrand is only "e^-x". For the second one you need to
do integration by parts:

INT of (udv) = uv - INT of (vdu)

It should be easy. Hint: choose u=x and dv=(e^-x)dx.

I hope this helps, but please let me know if you're still stuck (try to tell me exactly where).

Cheers,
David

David Montiel

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