Blog: `y = x, y = xe^(1 - (x/2))` Use a computer algebra system to find the exact volume of the solid obtained by rotating the region bounded by the...

Sunday, September 27, 2015

$\displaystyle{y}={x},{y}={x}{{e}}^{{{1}-{\left(\frac{{x}}{{2}}\right)}}}$ Use a computer algebra system to find the exact volume of the solid obtained by rotating the region bounded by the...

These curves are intersected at $\displaystyle{x}={0}$ and $\displaystyle{x}={2}.$

Between these points $\displaystyle{0}\leq{x}\leq{x}{{e}}^{{{1}-\frac{{x}}{{2}}}}\leq{2}.$

Let's use the method of rings.

The parameter of a ring is $\displaystyle{x}$ between $\displaystyle{0}$ and $\displaystyle{2}.$

The area of a ring is $\displaystyle\pi{\left[{{\left({3}-{x}\right)}}^{{2}}-{{\left({3}-{x}{{e}}^{{{1}-\frac{{x}}{{2}}}}\right)}}^{{2}}\right]}.$

Therefore the volume is equal to

$\displaystyle\pi{\int_{{0}}^{{2}}}{\left[{{\left({3}-{x}\right)}}^{{2}}-{{\left({3}-{x}{{e}}^{{{1}-\frac{{x}}{{2}}}}\right)}}^{{2}}\right]}{\left.{d}{x}\right.}.$

Computer algebra system WolframAlpha says that the exact value is $\displaystyle\frac{{{2}\pi}}{{3}}{\left({36}{e}-{3}{{e}}^{{2}}-{71}\right)}.$

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Sunday, September 27, 2015

$\displaystyle{y}={x},{y}={x}{{e}}^{{{1}-{\left(\frac{{x}}{{2}}\right)}}}$ Use a computer algebra system to find the exact volume of the solid obtained by rotating the region bounded by the...

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