Question
use integration to find volume of the cone below with height of 20 units and base radius of 5 units.
Answers
Bot
Volume of a cone = (1/3)πr2h
Volume = (1/3)π(5)2(20)
Volume = (1/3)π(25)(20)
Volume = (1/3)π(500)
Volume = 166.67π units3
Volume = (1/3)π(5)2(20)
Volume = (1/3)π(25)(20)
Volume = (1/3)π(500)
Volume = 166.67π units3
oobleck
the cone is formed by rotating the line y = 20-4x about the y-axis. So, using shells of thickness dx, we have
v = ∫[0,5] 2πrh dx
where r=x and h=y = 20-4x
v = ∫[0,5] 2πx(20-4x) dx = 500π/3
or, using discs of thickness dy,
v = ∫[0,20] πr^2 dy
where r=x = 5-y/4
v = ∫[0,20] π(5-y/4)^2 dy = 500π/3
v = ∫[0,5] 2πrh dx
where r=x and h=y = 20-4x
v = ∫[0,5] 2πx(20-4x) dx = 500π/3
or, using discs of thickness dy,
v = ∫[0,20] πr^2 dy
where r=x = 5-y/4
v = ∫[0,20] π(5-y/4)^2 dy = 500π/3
Damon
integral from y = 0 to y = h of pi r^2 dy
where r = y/4 (doing it upside down to simplify, not using r = 5 - y/4)
= integral from y = 0 to y = h of pi y^2/16 dy
= pi (1/16) h^3/3 = pi ( 8000 / 48) = 167pi
check
(1/3) pi r^2 h = (1/3) pi (25) 20 = 167 pi
where r = y/4 (doing it upside down to simplify, not using r = 5 - y/4)
= integral from y = 0 to y = h of pi y^2/16 dy
= pi (1/16) h^3/3 = pi ( 8000 / 48) = 167pi
check
(1/3) pi r^2 h = (1/3) pi (25) 20 = 167 pi