Question

Find the area between the curves:
y = x^3 - 9 x^2 + 14 x
and y = -x^3 + 9 x^2 - 14 x

Answer 1

Hi,
\(\Large\bf \color{royalblue}{\text{Welcome to OpenStudy! :)}}\)

Answer 2

hi

Answer 3

\[\Large\bf\sf y_1=x^3-9x^2+14x\]\[\Large\bf\sf y_2=-x^3+9x^2-14x\]Messing with y2 a little bit,\[\Large\bf\sf y_2=-(x^3-9x^2+14x) \qquad\implies\qquad y_2=-y_1\]

Answer 4

So ummmm

Answer 5

Do you understand how to find points of intersection?
(This will tell us from where and to where we're integrating).

Answer 6

yes make them equal to one another
\

Answer 7

Ok good.
In doing so, you should get something like:\[\Large\bf\sf x^3-9x^2+14x\quad=\quad 0\]Which gives us 3 intersections:
x=0, x=2, x=7

Answer 8

right

Answer 9

This one is a little tricky since we have 3 intersections.
Here's the graph just to give you an idea of what we're dealing with:
https://www.desmos.com/calculator/tzok9densw

We'll have to setup 2 integrals.
One from 0 to 2 and another from 2 to 7.
And then add the integrals afterwards.

Answer 10

i did!

Answer 11

and i am still getting it wrong!

Answer 12

\[\Large\bf\sf \int\limits_0^2 y_1-y_2\;dx\quad+\quad\int\limits_2^7y_2-y_1\;dx\]

Answer 13

So for our first integral:\[\Large\bf\sf \int\limits_0^2 x^3-9x^2+14x-(-x^3+9x^2-14x)\;dx\]

Answer 14

\[\Large\bf\sf =2\int\limits_0^2 x^3-9x^2+14x\;dx\]

Answer 15

shouldnt 14x be 28 x

Answer 16

14x--14x=28 x right

Answer 17

oh Nm..you divided by 2

Answer 18

Yah I just noticed that we have 2 of everything :D factored a 2 out.

Answer 19

right cool beans

Answer 20

now we do the anti deriv. right

Answer 21

Well that just takes care of the first region.
We'll have to do the other integral as well.

But ya let's go ahead and do the integration for this first one.

Answer 22

which is \[((x^4)/4)-3x^3+7x^2\]

Answer 23

did you get 16 as the answer for the first integral

Answer 24

yes good.

Answer 25

For the other integral, the order switches right?\[\Large\bf\sf \int\limits_2^7 (-x^3+9x^2-14x)-(x^3-9x^2+14x)\;dx\]

Answer 26

right

Answer 27

\[\Large\bf\sf =\quad -2\int\limits_2^7 x^3-9x^2+14x\;dx\]

Answer 28

2588.25?

Answer 29

bahh i dunno i didn't check it yet >.<

Answer 30

Hmm it should work out to 187.5 I think.

Answer 31

total?

Answer 32

\[\Large\bf\sf -2\left[\frac{1}{4}x^4-3x^3+7x^2\right]_2^7\]

Answer 33

Just for this piece, unless I punched something in wrong :o

Answer 34

could be me ill do it again

Answer 35

187.5?

Answer 36

i got 203.5 total and i was right!

Answer 37

Yayyy good job \c:/

Answer 38

Tough problem cause of those two areas D:

Answer 39

hey could i ask you antoher question or do i have to open up a new question thing again

Answer 40

nm ill open another one and close this one thnx

Answer 41

ya thats a good idea :D
I'm kinda multitasking right now, so if i can't get to it it's good to have a fresh window so others can see it.