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anonymous

  • one year ago

find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis y =(x^2)+1 , y = 9 - (x^2) about y = -1

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  1. freckles
    • one year ago
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    I like a little visual. |dw:1443749174250:dw|

  2. freckles
    • one year ago
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    find the x-coordinates of the intersections there for the limit part

  3. anonymous
    • one year ago
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    \[\pi[\int\limits_{-2}^{2}(10-x^2)^2-\int\limits_{-2}^{2}(x^2+2)^2\]

  4. anonymous
    • one year ago
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    this is what i have so far

  5. anonymous
    • one year ago
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    i think its wrong because i keep getting the wrong answer

  6. freckles
    • one year ago
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    |dw:1443749494904:dw| your radi look right

  7. freckles
    • one year ago
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    let me check your x-coordinates of your interesections

  8. freckles
    • one year ago
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    \[x^2+1=9-x^2 \\ 2x^2=8 \\ x^2=4 \\ x= \pm 2\]

  9. freckles
    • one year ago
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    what did you get as the answer after integration ?

  10. anonymous
    • one year ago
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    the final answer or do you want me to integrate it?

  11. anonymous
    • one year ago
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    ill integrate

  12. freckles
    • one year ago
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    give the answer and I will check it then if the answer is not what I have I might ask you to show me the integration

  13. anonymous
    • one year ago
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    i got 864pi for the final answer but wolframalpha says 256pi

  14. freckles
    • one year ago
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    ha so it is the integration

  15. freckles
    • one year ago
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    \[(10-x^2)^2=100-20x^2+x^4 \\ (2+x^2)^2=4+4x^2+x^4\] you get this for the expansion part?

  16. freckles
    • one year ago
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    the difference would be ... \[96-24x^2 \] \[\pi \int\limits_{-2}^{2} (96-24x^2) dx \\ \pi( 96x-8x^3)|_{-2}^{2}\]

  17. anonymous
    • one year ago
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    \[i get 100x-\frac{ 20x^3 }{ 3 }+\frac{ x^5 }{ 5 }-\frac{ x^5 }{ 5 }-\frac{ 4x^3 }{ 3 }+4x\]

  18. freckles
    • one year ago
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    I think that one term should be -4x

  19. freckles
    • one year ago
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    not +4x

  20. anonymous
    • one year ago
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    oh yes i see that

  21. freckles
    • one year ago
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    but anyways you should be able to combine like terms after making that correct and get what I have above

  22. freckles
    • one year ago
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    it is fine what you did integrate then simplify I prefer to simplify then integrate most of the time

  23. anonymous
    • one year ago
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    \[[100(2)-\frac{ 20(2)^3 }{ 3 }-\frac{ 4(2)^3 }{ 3 }-4(2)]-[100(-2)-\frac{ 20(-2)^3 }{ 3 }-\frac{ 4(-2) }{ 3 }-4(-2)]\]

  24. freckles
    • one year ago
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    ick why not clean up before pluggin in numbers

  25. freckles
    • one year ago
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    if you prefer this, this is fine... but it looks gross to me :p

  26. anonymous
    • one year ago
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    youre right but ive been working on this problem all day so im tryin to be explicit

  27. anonymous
    • one year ago
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    ok i got it thank you

  28. freckles
    • one year ago
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    \[[200-\frac{160}{3}-\frac{32}{3}-8]-[-200+\frac{160}{3}+\frac{32}{3}+8] \\ (200+200)+(-\frac{160}{3}-\frac{160}{3})+(\frac{-32}{3}-\frac{32}{3})+(-8-8) \\ (400)+(\frac{-320}{3})+(\frac{-64}{3})+(-16) \\ \\ (400-16)+(\frac{-320}{3}+\frac{-64}{3}) \\ (384)+(\frac{-384}{3}) \\ 384(1-\frac{1}{3}) \\ 384(\frac{2}{3}) \\ \frac{384}{3}(2) \\ (128)(2) \\ 256\]

  29. anonymous
    • one year ago
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    my math was just a little off

  30. freckles
    • one year ago
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    is the way I would have continued from your last step there

  31. freckles
    • one year ago
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    and of course shove the pi in

  32. anonymous
    • one year ago
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    i got all the integration and the equation and messed up on the algebra

  33. freckles
    • one year ago
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    I think if you have simplified before integration would have made it tons easier

  34. anonymous
    • one year ago
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    il keep that in mind these problems are taking longer and longer

  35. freckles
    • one year ago
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    \[\int\limits _{-2}^{2}[(10-x^2)^2-(x^2+2)^2 ]dx\\ \int\limits_{-2}^2 [(100-20x^2+x^4)-(x^4+4x^2+4)] dx \\ \int\limits_{-2}^2 [(100-4)+(-20x^2-4x^2)+(x^4-x^4)] dx \\ \int\limits_{-2}^2 [96-24x^2] dx \\ 96x-24 \frac{x^3}{3} |_{-2}^2 \\ 96x-8x^3|_{-2}^2 \\ (96 \cdot 2 -8(2)^3)-(96 \cdot (-2)-8(-2)^3) \\ (192-64)-(-192+64) \\(192+192)+(-64-64) \\ 384+(-128) \\ 256\]

  36. freckles
    • one year ago
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    just in case you wanted to see how I would tackled the integration and the after part

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