Approached Area Between Two Parabolas

Question

I have two parabolas:

$$y = x^2$$ $$y = x^2 + 1$$

From my understanding, the graphs would never cross, but they must get very close together when $x$ is very large. Hence what does the area approach to between the two curves? Or is it infinity?

As you can see by the image from Desmos, the bulk of the area is located when $x$ is small, the larger $x$ gets the less area is added to the total, so it must approach some number right?

This is just what I was thinking about whilst I was doing parabolas in further maths - I hope there's an interesting answer to it though :)

Answer 1

For each $x$, the two curves are exactly one unit apart in the $y$ direction. So the area between the curves over the interval $[a,b]$ is exactly $b-a$.

If you let $b\to\infty$ or $a\to-\infty$, the area is unbounded.

Answer 2

Hint:

The vertical distance between the two curves is: $y_2-y_1=x^2+1-x^2=1$ , for all the values of $x$.

So the area $A$ between the two curves is infinite: $$ A=\int_{-\infty}^{+\infty}(y_2-y_1)dx=\int_{-\infty}^{+\infty}dx $$