Hey guys, really need help with a Volumes question.
The volume enclosed between the curves y = x^4 and y = x^3 is rotated about the y-axis. Find the volume swept out using cylindrical shells
I'm getting a negative answer for some reason...
I'm doing Volume of y=x^4 revolved minus the volume of y=x^3 revolved.
delta(V) = 2*pi*r*h*delta(r).
For y=x^4, r = x, h = x^4, so V = Integral from 0 to 1 of 2*pi*x^5
For y=x^4, r = x, h = x^3, V = Integral from 0 to 1 of 2*pi*x^4
Volume of region enclosed = Integral from 0 to 1 of [2*pi*x^5 - 2*pi*x^4]
I'm getting a negative number
Stacey Warren - Expert brainly.com
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So it should be
The area enclosed between y=x^4 and the line y = 1, rotated about the y-axis, MINUS The area enclosed between y=x^3 and the line y = 1 rotated about the y-axis
I know taking the absolute value would be the logical thing to do, but I really don't see why the answer is negative. I'm subtracting a smaller area from a larger area. I should be getting a positive answer without having to absolute value it
x^4 is under x^3 between 0 and 1; so it simply means you subtracted the lesser from the greater
and i proved that by:
(1/2)^4 < (1/2)^3
1/16 < 1/8
and when you subtract a lesser from a greater; it gives you a negative...
Ah, I see what happened now.
This is what confused me: if you imagine the solids formed when the curves are rotated about the y-axis, the one formed by y=x^4 is bigger, isn't it?
(3/4)^4 < (3/4)^3
81/256 < 27/64
x^3 is bigger then x^4 between 0 and 1
I see that the y-values are bigger, but I really don't see how the volume is bigger.
Oh well... I don't see it, but it makes sense. Thank you!
spose this was the volume of a box with a smaller box inside...
5 cubic feet - 4 cubic feet = 1 cubic feet left over
but if we do the opposite
4 cubic feet - 5 cubic feet = -1 cubic feet its the same |absvalue|
we simply get a - value rather than a positive value whih indicates that we simply reversed the numbers