Find the volume of revolution of the region generated,

bounded by graph $\displaystyle y = 6 - 2x -x^2$

and the indicated line $\displaystyle y = x + 6.$

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- Apr 3rd 2009, 04:27 PMShyamVolume of Revolution
Find the volume of revolution of the region generated,

bounded by graph $\displaystyle y = 6 - 2x -x^2$

and the indicated line $\displaystyle y = x + 6.$ - Apr 3rd 2009, 04:34 PMskeeter
- Apr 3rd 2009, 05:42 PMShyam
- Apr 3rd 2009, 06:34 PMShyam
here's the work I did:

After solving the equations of parabola and line, I found their point of intersection.

The point of intersection of parabola and line are (-3, 3) and (0, 6)

$\displaystyle V = \pi \int\limits_{ - 3}^0 {\left[ {\left( {6 - 2x - x^2 } \right)^2 - \left( {x + 6} \right)^2 } \right]} .dx \hfill \\$

$\displaystyle = \pi \int\limits_{ - 3}^0 {\left[ {\left( {x^4 + 4x^3 - 8x^2 - 24x + 36} \right) - \left( {x^2 + 12x + 36} \right)} \right]} .dx \hfill \\$

$\displaystyle = \pi \int\limits_{ - 3}^0 {\left[ {x^4 + 4x^3 - 9x^2 - 36x} \right]} .dx \hfill \\$

$\displaystyle = \pi \left[ {\frac{{x^5 }}

{5} + x^4 - 3x^3 - 18x^2 } \right]_{ - 3}^0 \hfill \\$

$\displaystyle = \pi \left[ {\left( {\frac{{\left( { - 3} \right)^5 }}

{5} + \left( { - 3} \right)^4 - 3\left( { - 3} \right)^3 - 18\left( { - 3} \right)^2 } \right) - \left( {0 + 0 - 0 - 0} \right)} \right] \hfill \\$

$\displaystyle = \pi \left[ {\frac{{ - 243}}

{5} + 81 + 81 + 162} \right] \hfill \\$

$\displaystyle = \pi \left[ {\frac{{ - 243}}

{5} + 324} \right] \hfill \\$

$\displaystyle = \frac{{1377\pi }}

{5} \hfill \\ $

This answer seems too big to me. Is it correct? - Apr 4th 2009, 07:28 AMskeeter
what you have shown is the work required for a rotation of the region about the x-axis, not about the line y = x+6.

this is correct ...

Quote:

$\displaystyle = \pi \left[ {\frac{{x^5 }}

{5} + x^4 - 3x^3 - 18x^2 } \right]_{ - 3}^0 \hfill \\$

Quote:

$\displaystyle = \pi \left[ {\left( {\frac{{\left( { - 3} \right)^5 }}

{5} + \left( { - 3} \right)^4 - 3\left( { - 3} \right)^3 - 18\left( { - 3} \right)^2 } \right) - \left( {0 + 0 - 0 - 0} \right)} \right] \hfill \\$

Quote:

$\displaystyle = \pi \left[ {\frac{{ - 243}}

{5} + 81 + 81 + 162} \right] \hfill \\$