How do you use the disk method to find the volume of the solid formed by rotating the region bounded

Kane Barr

Kane Barr

Answered question

2022-06-04

How do you use the disk method to find the volume of the solid formed by rotating the region bounded by y=2x and y = x 2 about the y-axis?

Answer & Explanation

kilometrocerobiy4v

kilometrocerobiy4v

Beginner2022-06-05Added 3 answers

y=2x and y = x 2 intersect at (0,0) and at (2,4)
I can't get both curves on one graph, but I'll assume you can graph the line and the parabola.
Rotating around the y axis and using disks, means our independent variable will be y.
The representative slice is horizontal with left end (little radius: r) on the line
x = 1 2 y and right end (big radius: R) on the parabola x = y
The thickness of the disk will be dy, and the limits of integration will be 0 to 4
0 4 π R 2 π r 2 d y = π 0 4 ( ( y ) 2 ( y 2 ) 2 ) d y
π 0 4 ( y y 2 4 ) d y = π [ y 2 2 y 3 12 ] 0 4 = π [ 8 16 3 ] = 8 π 3

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