i have two parabolas x=y^2-1 and x=2y^2-2
i have to find the area between them using multiple integrals
i know by simple integrals but how to do it by multiple integrals
please help
Start by drawing a sketch. You'll see that the region is easiest evaluated using horizontal strips, with each strip bounded on the left by $\displaystyle \displaystyle \begin{align*} x = 2y^2 - 2 \end{align*}$ and on the right bounded by $\displaystyle \displaystyle \begin{align*} x = y^2 - 1 \end{align*}$. Then when summing all these strips, the strips are bounded below by $\displaystyle \displaystyle \begin{align*} y = -1 \end{align*}$ and bounded above by $\displaystyle \displaystyle \begin{align*} y = 1 \end{align*}$.
So the area is given by $\displaystyle \displaystyle \begin{align*} \int_{-1}^1{\int_{2y^2 - 2}^{y^2 - 1}{1\,dx}\,dy} \end{align*}$.
Another approach is to use the difference of these two functions, and integrate that. You have to use modulus (absolute value) of the difference, and divide integration interval depending on the sign of it. In that method, integration limits will be constants, not y^2 -1 and 2y^2 - 2.
Salahuddin
Maths online