Finding the area between 3 curves

  • Thread starter xstetsonx
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  • #1
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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!!!
 

Answers and Replies

  • #2
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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 in to a true triangle and a curved slice. Find the areas of each separately.
 
  • #3
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sorry still don't know what to do. my book doesn't have any examples like this question
 
  • #4
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Look at this image:

4059633926_f5e8fc5039_m.jpg


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

System 1
[tex]y=3x^3-3x[/tex]
[tex]y=3x[/tex]

A(x,y) =

System 2
[tex]y=3x^3-3x[/tex]
[tex]y=9-x[/tex]

B(x,y) =

System 3
[tex]y=9-x[/tex]
[tex]y=3x[/tex]

C(x,y) =

Now you can break it in to two differences of integrals.

One from A to B and another from B to C.
 
  • #5
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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
 
  • #6
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The first integral will be:

[tex]\int_{x_1}^{x_2}(x^3-3x) -\int_{x_1}^{x_2}3x[/tex]

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

[tex]\int_{x_2}^{x_3}(9-x) -\int_{x_2}^{x_3}3x[/tex]

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.
 
  • #7
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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
 
  • #8
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Yup. You'll get 3 solutions for that since the graphs intersect 3 times. Just pick the one with the largest value...
 
  • #9
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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
 
Last edited:
  • #10
270
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They should all be positive.
 

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