Vol = π∫(27x^3)^2 dx from 1 to 2
= π∫ 729x^6 dx from 1 to 2
= π[729/7 x^7] from 1 to 2
= π[ 729/7*128 - 729/7*1]
= ....
your turn
Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.
y = 27x3, y = 0, x = 1; about x = 2
3 answers
oops, did not read the questions carefully
I simply rotated everything about the x-axis, my mistake.
I simply rotated everything about the x-axis, my mistake.
y = 27x^3
y/27 = x^3
y^(1/3) / 3 = x
The whole solid rotated around the line x = 2 , from y = 0 to y = 27 is
π∫ (2 - x)^2 dy from 0 to 27 , ..... (when x = 1, y = 27)
= π∫ (2 - (1/3)y^(1/3) )^2 dy
= π ∫( 4 - (4/3)y^(1/3) + (1/9)y^(2/3) ) dy
= π[ 4y - y^(4/3) + (1/15)y^(5/3) ] from 0 to 27
= π ( 108 - 81 + 243/15 - 0)
= 216π / 5
but that includes the "cylinder" from x = 1 to 2
which is π(2-1)^2 = π
final volume = 216π/5 - π = 211π/5 cubic units
check my arithmetic
y/27 = x^3
y^(1/3) / 3 = x
The whole solid rotated around the line x = 2 , from y = 0 to y = 27 is
π∫ (2 - x)^2 dy from 0 to 27 , ..... (when x = 1, y = 27)
= π∫ (2 - (1/3)y^(1/3) )^2 dy
= π ∫( 4 - (4/3)y^(1/3) + (1/9)y^(2/3) ) dy
= π[ 4y - y^(4/3) + (1/15)y^(5/3) ] from 0 to 27
= π ( 108 - 81 + 243/15 - 0)
= 216π / 5
but that includes the "cylinder" from x = 1 to 2
which is π(2-1)^2 = π
final volume = 216π/5 - π = 211π/5 cubic units
check my arithmetic
2 answers