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

- anonymous

- Stacey Warren - Expert brainly.com

Hey! We 've verified this expert answer for you, click below to unlock the details :)

- katieb

I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!

- amistre64

where do x^4 and x^3 match up? 0 and 1 right?

- amistre64

try again:
2pi {S} x(x^4) - x(x^3) dx ; [0,1]

- amistre64

x^5 ints to x^6/6 and x^4 ints up to x^5/5

Looking for something else?

Not the answer you are looking for? Search for more explanations.

## More answers

- amistre64

2pi/6 - 2pi/5 = ?

- anonymous

What's the {S}?

- amistre64

its the symbol I use to type in the integration elongated 's'

- anonymous

Ah, okay ^^
but doesn't 2pi/6 - 2pi/5 = -pi/15?

- amistre64

10pi - 12pi
---------- = -2pi/30 = -pi/15 yes; which simply means that
30 you got the numbers backwards
watch tho:
5-4 = 1
4-5 = -1
so just disregard the sign

- amistre64

take the absolute value |-pi/15|

- anonymous

But when graphed, between 0 and 1, y = x^4 is the one with the larger volume, no? So why would it be backwards?

- amistre64

x^4 is flatter than x^3; so its actually below it
try .5^4 and .5^3 for trial

- amistre64

.5^4 is actually a smaller number than .5^3 i think

- amistre64

.625 and .125 right?

- anonymous

but 5>1.
between 0 and 1, y=x^4 is below y=x^3, which means y=x^4 has a larger volume. So I shouldn't be getting a negative number when done in that order

- amistre64

.0625 and .125 maybe

- amistre64

0.0625 and 0.125 the ^4 is less than the ^3

- amistre64

since we are working with numbers that are fractions; we get x^4 under x^3

- amistre64

anything greater than 1 reverses it

- anonymous

That's the graph I'm working with. The yellow highlighted area is what I'm revolving.

##### 1 Attachment

- anonymous

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

- anonymous

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

- amistre64

##### 1 Attachment

- amistre64

x^4 is under x^3 between 0 and 1; so it simply means you subtracted the lesser from the greater

- amistre64

and i proved that by:
(1/2)^4 < (1/2)^3
1/16 < 1/8

- amistre64

and when you subtract a lesser from a greater; it gives you a negative...

- anonymous

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?

- amistre64

(3/4)^4 < (3/4)^3
81/256 < 27/64

- amistre64

x^3 is bigger then x^4 between 0 and 1

- anonymous

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!

- amistre64

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|

- amistre64

we simply get a - value rather than a positive value whih indicates that we simply reversed the numbers

- anonymous

Ah! Okay, I get it now! ^^ Thanks so much!

Looking for something else?

Not the answer you are looking for? Search for more explanations.