Finding the area between 3 curves

[xstetsonx]

fx=3x^3-3x, gx=3x, and hx=9-x. Find the area

I kown top - bottom and right - left. but in here i am not sure what to do and what the boundaries are. can some one show me the work how to do this problem? i am kinda confuse how to do this kind of problem with 3 curves. THANK YOU!

 

[futurebird]

Did you draw a picture? The region looks like a triangle, but the side formed by 3x^3-3x is curved. I would start by fining the verticies of the triangle... or the points where these lines intersect.

So we need to solve 3 systems of equations. Once you have done that you can divide the region into a true triangle and a curved slice. Find the areas of each separately.

 

[xstetsonx]

sorry still don't know what to do. my book doesn't have any examples like this question

 

[futurebird]

Look at this image:

First find the points where the graph intersects to form the triangle by solving these three systems of equations:

System 1
$$y=3x^3-3x$$
$$y=3x$$

A(x,y) =

System 2
$$y=3x^3-3x$$
$$y=9-x$$

B(x,y) =

System 3
$$y=9-x$$
$$y=3x$$

C(x,y) =

Now you can break it into two differences of integrals.

One from A to B and another from B to C.

 

[xstetsonx]

ok i graphed and i found the 3 points. A:(1.4142,4.2426) B:(1.5958,7.4042) C:(2.25,6.75)
so when u say break it into 2 parts you mean (∫1.5958 on top, 1.4142 bottom (9-x)-(3x) dx)? am i heading to the right direction? if i am i don't know how to get the other part

 

[futurebird]

The first integral will be:

$$\int_{x_1}^{x_2}(x^3-3x) -\int_{x_1}^{x_2}3x$$

Where x1 and x2 are the x values from the points A and B. then add that to the 2nd integral:

$$\int_{x_2}^{x_3}(9-x) -\int_{x_2}^{x_3}3x$$

Where x2 and x3 are the x values from the points B and C.

Don't just take my word for it! Make certian you understand *why* --think about the region each integral represents, shade in the graph if needed.

The values you found look reasonable, but it looks like you used a graphing calculator? If I were teaching this course I'd want an exact value in radicals. Just check that your prof. is OK with aprox. values.

 

[xstetsonx]

HA. i see. and to get the exact radical you just set 9-x=3x^3-3x to get B and so on right? thank you so much btw

 

[futurebird]

Yup. You'll get 3 solutions for that since the graphs intersect 3 times. Just pick the one with the largest value...

 

[xstetsonx]

oops wait a minute why do i get a negative value when i do ∫ x^3-3x? and the bonds are 1.5958 and 1.4142 right? shoulden't they all be positive?

nvm my mistake

 

[futurebird]

They should all be positive.