Find the area of the region between the curves y=12-x^2 and y=x^2-6

Question

amistre64 · Answer

height times width = area right?

anonymous · Answer

Yes

amistre64 · Answer

y1-y2 = distance between them, and dy = width

anonymous · Answer

But i dont know how to use it in this equations. im confused

amistre64 · Answer

$$\int (12-x^2)-(x^2-6)dx$$

amistre64 · Answer

i shoulda said dx to begin with but my mind was a bit off :)

amistre64 · Answer

only thing missing is how far to go

anonymous · Answer

so integrate this and then what?

amistre64 · Answer

then apply the limits that need to be calculated

amistre64 · Answer

when does 12-x^2 =x^2-6 ??

amistre64 · Answer

2x^2 = 18

x^2 = 9

x = +- 3

anonymous · Answer

so it'll that integral from -3 to 9

amistre64 · Answer

$$\int_{-3}^{3} (12-x^2)-(x^2-6)dx$$or simply$$2\int_{0}^{3} (12-x^2)-(x^2-6)dx$$

anonymous · Answer

oh ok i understand ...thank you.

anonymous · Answer

so i just have to solve that integral?

amistre64 · Answer

your welcome; and it might come ount negative, but that just means we subtracted in a mismatched order; ignore any signage in the answer and youll be fine

amistre64 · Answer

the books always say:  subtract the bottom curve from the top; but it really doenst matter in the end.  One answer might be 9 while the other -9; so just ignore the sign

anonymous · Answer

thanks, do i do the same for Find the area of the region between the curves y=x^2 and y^2=x

amistre64 · Answer

yes, but you would do best to get them speaking the same terms;

sqrt(y) = x  and y^2 = x  for example

amistre64 · Answer

|dw:1318558541442:dw|

anonymous · Answer

sqrt(y).. where did that come from?

amistre64 · Answer

really?

y = x^2  ; sqrt each side

sqrt(y) = x

anonymous · Answer

lol oh, sorry.. totally forgot

amistre64 · Answer

i know i know, you know how much algebra i had to forget to learn trig :)

anonymous · Answer

Lol, this stuff is totally confusing. so i would integrate x^2-x?

amistre64 · Answer

that wouldnt be to practical; they are speaking different terms at the moment so are hard to reconcile.  kinda like adding unlike denominators on fractions; we have to alter them to speak the same language

amistre64 · Answer

y=x^2   and   y^2=x

y = x^2         y = sqrt(x)

int (x^2 - sqrt(x)) from 0 to 1

amistre64 · Answer

now that will turn out a negative answer; so ignore the sign