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AverageJoe20

  • 3 years ago

Fine the volume of the solid in which the solid lines between planes perpendicular to the y-axis at y=0 and y=2. The cross sections perpendicular to the y-axis are circular disks with diameters running from the y-axis to the parabola x= sort(5)y^2.

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  1. TuringTest
    • 3 years ago
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    sorry, is the function is\[x=\sqrt5y^2\]???

  2. AverageJoe20
    • 3 years ago
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    Yep!

  3. TuringTest
    • 3 years ago
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    |dw:1360708102653:dw|here is the graph, now we need to consider discs coming out in the z-direction. Each disk will have an area of \(\pi r^2\). The radius of each disk will be half the distance from the y-axis to the parabola, so we will use that for a formula for the area as a function of y.

  4. AverageJoe20
    • 3 years ago
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    Yeah I get to that, it is just setting up the integral that is messing me up.

  5. TuringTest
    • 3 years ago
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    What do you have for the function for the area of each disk? that will be your integrand. The region is bound by the planes y=0 and y=2, so if you integrate with respect to y (which you should do) those are your bounds. Are you still stuck? If so, where?

  6. AverageJoe20
    • 3 years ago
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    So would it be from 0 to 2 of the integral \[\sqrt{5}y ^{2} dy \] ?

  7. TuringTest
    • 3 years ago
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    yes to the bounds, no to the integrand. that is the distance from the y-axis to the parabola, not the area of each disk.|dw:1360708680890:dw|so the radius of each disk is half the distance to the parabola\[r=\frac{\sqrt5}2y^2\]now use the formula for the area of a circle\[A=\pi r^2\]what do you get?

  8. AverageJoe20
    • 3 years ago
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    \[\left(\begin{matrix}5 \\4\end{matrix}\right)pi y^4?\]

  9. AverageJoe20
    • 3 years ago
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    I know I have to take the integral though....

  10. TuringTest
    • 3 years ago
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    yes, so just integrate that with respect to y, evaluate the given bounds and you're done.

  11. AverageJoe20
    • 3 years ago
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    I love you.

  12. AverageJoe20
    • 3 years ago
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    Thank you sooo much! This problem has been tripping me up and you made me look dumb haha. Thanks for the help!

  13. TuringTest
    • 3 years ago
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    Happy to help :D

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