Area between two curves that do not intersect

Question

the curves are $x^2 = 4y$ and $x^2=4y-4$ these are just the same parabolas but the other one is shifted up by one unit.

I have been thinking of 3 possibilities that might be the answer.

1. The area is equal to infinite sq. units

2. The area is equal zero

3. The area is undefined

The answer I have concluded that is probably the most correct is that the area is undefined because:

1. The area is not enclosed by the two curves

2. Infinity is not a number

3. The area is definitely not zero since the curves are not overlapping

So my question is if I answered this correctly.

Answer 1

You can rewrite this as an improper integrals:

$$\int_{-\infty}^\infty \left({x^2 + 4\over 4} - {x^2\over 4}\right) \,\mathrm dx = \int_{-\infty}^\infty 1\,\mathrm dx$$

It becomes obvious this does not converge thus the area is not finite:

$$\lim_{b\to\infty} x \,\Big|^b_{-b} \rightarrow \infty - (-\infty) = \infty$$

The notion that the area is undefined because the curves do not cross is wrong. Consider the Gaussian Integral which is between the functions $f(x) = e^{-x^2}$ and $g(x) = 0$. They do not cross, yet the integral from negative infinity to positive infinity is finite:

$$\int_{-\infty}^\infty e^{-x^2} \,\mathrm dx = \sqrt{\pi}$$

Note that $g(x) = 0$ is technically a curve. Mathematically speaking, a curve is a generalization of a line.

Answer 2

You really got a improper integral for the area between those two curves. $\text{Area} = \displaystyle \int_{-\infty}^{\infty}\left(\dfrac{x^2+4}{4}-\dfrac{x^2}{4}\right)dx=\displaystyle \lim_{a\to \infty}\displaystyle \int_{-a}^{a}1dx=+\infty$

Answer 3

While the other answers tell you why the first option is the correct answer, let me explain where your arguments and conclusions are correct or wrong.

While your first argument is a bit fishy, the second is true but the conclusion is wrong. And the third argument is correct to exclude the 3. option. Therefore you concluded wrongly the third option instead of the correct first option.

But here a bit more details to your arguments:

1. The area is not enclosed by the two curves

The area is enclosed by the two curves. You might have the feeling that there are missing a part of the boundary, because the curves doesn't intersect, but a boundary doesn't need to go around the area in one draw. So, infinite large areas are possible too.

2. Infinity is not a number

That's true and very important to know. But in fact, the measure of an area maps an area to a real number or infinity. Hence, infinity is a valid measure of an area.

3. The area is definitely not zero since the curves are not overlapping

That is correct. That's why the measure of the decribed area has to be positive.