How do you find the volume of the solid obtained by rotating the region bounded by the curves y

gaiaecologicaq2 2022-07-01 Answered
How do you find the volume of the solid obtained by rotating the region bounded by the curves y = 2 x 2 + 5, and y=x+3 and the y-axis, and x=3 rotated around the x axis?
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Answers (1)

Kayley Jackson
Answered 2022-07-02 Author has 16 answers
Sketch the region. Note that y = 2 x 2 + 5 is above (greater than) x+3, so the parabola is farther from the axis of rotation.
Therefore:
At a particular x, the large radius is, R = 2 x 2 + 5, and the small radius is
r=x+3. The thickness of the disks is dx
The volume of each representative disk would be π r a d i u s 2 t h i k n e s s. So the large disk has volume: π ( 2 x 2 + 5 ) 2 d x and the small one has volume π ( x + 3 ) 2 d x
The volume of the washer is the difference, or π R 2 d x π r 2 d x and the resulting solid has volume:
V = π 0 3 ( ( 2 x 2 + 5 ) 2 ( x + 3 ) 2 ) d x
= π 0 3 ( 4 x 4 + 19 x 2 6 x + 16 ) d x
You can finish the integral to get 1932 π 5
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