volumez question (1 Viewer)

[bleakarcher]

the region bounded by the curve y=e^x and the lines x=1 and y=1 is rotated about the x-axis. find the exact volume of the solid formed

 

[Hermes1]

the region bounded by the curve y=e^x and the lines x=1 and y=1 is rotated about the x-axis. find the exact volume of the solid formed

use cyclindrical shells

V = 2pi(x1-x2)(y-1)(change in y)
V = 2pi(1-log y)(y-1)...
get an integral with limits 1 and e. and solve. you will need to use integration by parts on some.

 

[darkccc]

This question requires a 3 step process.

1. the line y=1 intersects y=e^x at x=0
and y^2=e^(2x)
2. The cylinder made by the volume of x=1 and y=1 is (pi) because the cylinder has radius of 1, and "height" of 1.

3. Hence, the volume is = Integrate [1,0] pi*e^(2x)dx - pi
=pi(0.5e^2-0.5)-pi
=pi0.5e^2-1.5pi cubic units.

Sorry for the lack of symbols (my computer cant input complex mathematical symbols...)
I hope that helped

 

[Hermes1]

This question requires a 3 step process.

1. the line y=1 intersects y=e^x at x=0
and y^2=e^(2x)
2. The cylinder made by the volume of x=1 and y=1 is (pi) because the cylinder has radius of 1, and "height" of 1.

3. Hence, the volume is = Integrate [1,0] pi*e^(2x)dx - pi
=pi(0.5e^2-0.5)-pi
=pi0.5e^2-1.5pi cubic units.

Sorry for the lack of symbols (my computer cant input complex mathematical symbols...)
I hope that helped

sorry this method is better than mine. i got so used to using cylindrical shells i forgot the other method

 

[bleakarcher]

thanks arj7, i want to know y my answer was wrong also.
i said

let the volume of formed by rotating a typical strip of width (delta x) and height (y-1) lying within the region bounded by the curve y=e^x and lines x=1 and y=1 at the point P(x,y) about x-axis be (delta V)
(delta V)=2(pi)y[y-1]*(delta x)
from then i ended up wif an answer completely wrong. this has been stressing me out. wat have i done wrong?

 

[Comeeatmebro]

This question requires a 3 step process.

1. the line y=1 intersects y=e^x at x=0
and y^2=e^(2x)
2. The cylinder made by the volume of x=1 and y=1 is (pi) because the cylinder has radius of 1, and "height" of 1.

3. Hence, the volume is = Integrate [1,0] pi*e^(2x)dx - pi
=pi(0.5e^2-0.5)-pi
=pi0.5e^2-1.5pi cubic units.

Sorry for the lack of symbols (my computer cant input complex mathematical symbols...)
I hope that helped

^this

 

[bleakarcher]

i think that may be wrong now, i think it is

V=integral from x=1 to x=2pi*y[(1-log[e]y] *dy

 

[seanieg89]

 

[failwhale]

beast