SOLUTION: Find the first moment of the area under y = {e}^{−x} from x = 0 to x = 1 about the y axis.

Algebra ->  Test -> SOLUTION: Find the first moment of the area under y = {e}^{−x} from x = 0 to x = 1 about the y axis.      Log On


   

Question 857705: Find the first moment of the area under
y = {e}^{−x}
from x = 0 to x = 1 about the y axis.
Answer by rothauserc(4718) About Me  (Show Source):
You can put this solution on YOUR website!
We are given y = {e}^{−x} and asked for area about y axis for x = 0 to x = 1
first moment area about y axis = integral (from y= 1 to 1/e) x dy
take ln of both sides and solve for x
ln y = -x
x = - ln y
integral (from y= 1 to 1/e) (-ln y)dy
now you can solve by parts by taking -ln y as first function & 1 as 2nd function
let u = -ln(y)
let v = 1dy
which gives
du = -1/y dy
v = y
we have the formula
integral (from y= 1 to 1/e) (-ln y)dy = y * ln(y) - integral y * 1/y dy
since integral y * 1/y dy = integral dy = y + c
we get
integral (from y= 1 to 1/e) (-ln y)dy = -y*ln(y) -y +C
now evaluate each term from y = 1 to 1/e
note ln (1) is 0 and ln (1/e) = -1