Question

Find the area between the curves. x=(2y^2)-2, x-(y^2)=7
Please help!

Answer 1

Consider any shape, like a circle with a square cut out of it. How would you find the area of this donut-like shape? Just find the area of the square and the area of the circle and subtract out the area of the square!

Similarly, you will just integrate them both to find their areas and then subtract them!

Answer 2

You would want to find first the point of intersection of the two functions. To do this, you may substitute x= (2y^2) - 2 to the 'x' of your second expression. Then you'll have an equation like this (after transposing 7).
\[(2y^{2} - 2) - (y^{2} - 7) = 0\]
Solve for the zeros of the new function above for the intersections. The intersections of the two curves will serve as your limit of integration. I can sense that both of them is a parabola opening to the right. Determine the vertex of each parabola so you'll have an idea which is which. The graphs may look like this.

You integrate in terms of y so you add up areas of horizontal rectangular strips with width dy. The integral set-up will look like this
\[\int\limits_{a}^{b} [f(y)-g(y)]dy\]
where a and b are your limits of integration, f(y) is the function to the left, g(y) is the function to the right.
Solve the definite integral to find the area.