Area finite region bounded by the curves

[Petrus]

Hello MHB,
I got stuck on an old exam
determine the area of the finite region bounded by the curves \(\displaystyle y^2=1-x\) and \(\displaystyle y=x+1\) the integration becomes more easy if we change it to x so let's do it
\(\displaystyle x=1-y^2\) and \(\displaystyle x=y-1\)
to calculate the limits we equal them
\(\displaystyle y-1=1-y^2 <=> x_1=-2 \ x_2=1\)
so we take the right function minus left so we got
\(\displaystyle \int_{-2}^1 y-1-(1-y^2) <=> \int_{-2}^1 y+y^2-2\) and I get the result \(\displaystyle - \frac{9}{2}\) and that is obviously wrong... What I am doing wrong?

Regards,
\(\displaystyle |\pi\rangle\)

 

[MarkFL]

Over the limits of integration, the parabolic function is greater than the linear function, this is why it is a good idea to sketch the region first so that you can see more clearly what you need to do. :D

 

[Petrus]

Over the limits of integration, the parabolic function is greater than the linear function, this is why it is a good idea to sketch the region first so that you can see more clearly what you need to do. :D

Thanks alot! I learned a lesson this time :) I did do it in my brain and that was not cleaver! Thanks a lot for the fast responed!:) Now I get \(\displaystyle \frac{9}{2}\) that is correct :)

Regards,
\(\displaystyle |\pi\rangle\)