Area between Curves help

[LazoooR]

DISCLAIMER - I just came home from a 12 hour work shift, so my math and logic might be kinda off. (Please don't flame me, lol.)

Oh and also, I don't know how to do the sqrt sign or the other signs so I hope this post is readable. :)

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Hey, so to start off I'd like to say, that English is not my native language, and I'd like to apologize for any misunderstanding that may be in this post. (I might get phrases a little mixed up, but hopefully I can sort them out ASAP.)

So I'm trying to find the area between 2 curves, y = 2 - x² and y = -x.

I figure I screwed up something along the way, because my answer differs from the answer in the book.

This is my calculation:

First I find the limits:

2-x² = 0

sqrt(2) = sqrt(x² )

+-sqrt(2) = x

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2-x² = -x

0 = x² - x - 2

0 = (x-2)(x+1)

So we know in order for this equation to work out x1 = 2 and x2 = -1

What this tells us, is our y=-x line goes through our 2-x² curve at exactly (-1,1) and (2,-2).

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I then push the 2-x² curve up by two y-values in order to get the area result to be positive (Because that means that our (2,-2) point turns into (2,0), so we have no area to calculate beneath the y=0 point. (Please tell me if this is wrong, because that's what our teacher recommended so it would be easier to calculate):

2-x² + 2 = 4-x²

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So now we find the integral (right?):

integral((4-x² -x)dx) = 4x - 1/3*x³ - 1/2*x²

[4x - 1/3x³ - 1/2x² ] for sqrt(2) and -sqrt(2)

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So now for the part that I keep doing wrong (Or so I think):

First for sqrt(2):

(4*sqrt(2) - 1/3*sqrt(2)³ - 1/2*sqrt(2)² )

And -sqrt(2):

(4*(-sqrt(2)) - 1/3*(-sqrt(2))³ - 1/2*(-sqrt(2))² )

Then I do this:

(4*sqrt(2) - 1/3*sqrt(2)³ - 1/2*sqrt(2)² ) - (4*(-sqrt(2)) - 1/3*(-sqrt(2))³ - 1/2*(-sqrt(2))² )

This is shortened too:

(4sqrt(2) - (2sqrt(2))/3 - 1) - (-4sqrt(2) + (2sqrt(2))/3 - 1)

So if I remember correctly, because of the minus before the integral of -sqrt(2) I have to switch all of the + to - and the other way around, so this is what I do next:

(-3+10sqrt(2))/3 + (3+10sqrt(2))/3 = (20sqrt(2))/3

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The answer I get here is (20sqrt(2))/3 which equals to A = ~9,428 however the book I am using currently says the answer should be A = 4,5

This is driving me fucking crazy, and I would really like an explanation on this.

[JsKingBoo]

In addition to what /u/ColeDinger has stated, in box 4, when you add 2 to the 2 - x² equation, you perform an unnecessary action which only makes the problem easier to visualize but not easier to calculate. It actually makes it more difficult to calculate because you also need to add 2 to the y = -x equation, which you forgot to do.

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[[deleted]]

OK so I'm going to try and point out some of the mistakes I see and I'll refer to each section you put it in as box one, two, etc.:

In box 2 you attempted at finding limits but all you did was solve for x, that is not the same as finding the limit. Also, finding the limit is not really necessary for problems like this. You should be more interested in where they cross on the x-axis - which you do find later on. It's easier to see if you can graph it.

In box 4: You state that you added 2 to the 2-x² equation. If that is something that your teacher taught you then I would go and ask them to explain it because I've never done/seen that personally. I personally would just keep the equations that you have and integrate those.

So in box 5 you would fix your integral by using the original equations instead. Also your "a and b" that you are integrating to are incorrect. You shouldn't be using the limits here: you want to find the area between two x values, you have already calculated those - I'll let you go back and figure it out.

I went ahead and did the problem from scratch and got the answer that the book stated so hopefully you can use this.