Volume by rotating?

I've tried this so many times! Please Help

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  • Pope
    Lv 7
    1 month ago

    I certainly would not use integration for this one. I often lament that students are held back in calculus because they do not first master the fundamentals of geometry.

    Begin by rotating rectangle OABC about axis BC. That generates a right cylinder of radius 5 and length 1. Its volume is 25π.

    Now remove the solid generated by regions R₂ and R₃. Together they generate the right cone having the same radius and length as the cylinder above. The cone has 1/3 the volume of the cylinder, so its removal leaves 2/3 of it.

    (2/3)(25π) = 50π/3

  • rotchm
    Lv 7
    1 month ago

    Using disc method (washer) .

    At x=x, draw a vertical rectangle in R1. Its height is thus 5x, and width dx.

    Rotating it makes a washer of volume dV = π(5² - (5-5x)²) dx. Convince yourself of this; its just highschool geometry.

    Now, just integrate form 0 to 1.

    Done!

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