Came across a problem on social media,

Find the area of the region bounded by a parabola, $y = x^2 + 6$ and line a line $y = 2x + 1$.

I tried to draw it on paper and they didn't seem to intersect. So I drew them online (attached screenshot). My answer was 0, but someone said that we assume they meet at infinity and answer would be infinity. Parallel lines don't diverge like these do, so I think we can assume that they would never interest at infinity.

enter image description here

  • $\begingroup$ I think so, which means that the original question is stated wrong $\endgroup$ Jul 24, 2020 at 21:02
  • 1
    $\begingroup$ The originally intended question may also have additional restrictions such as "where $x$ ranges from to $0\leq x \leq 5$" in which case the question is perfectly well defined. You say you saw this posted on social media... well... a lot of people who share problems on social media do not understand them well enough to repeat the problem correctly and so details are often forgotten or typos or other slight changes to the problem occur. I wouldn't worry too much about this. $\endgroup$
    – JMoravitz
    Jul 24, 2020 at 21:28

1 Answer 1


$$x^2 + 6 = 2x + 1$$ $$x^2 - 2x + 5$$ $$\frac{2 \pm \sqrt{4 - 4(5)}}{2}$$

As you can see by analyzing the discriminant, this quadratic has no real roots, so there are no points at which the two curves intersect. You could say that the area between the curves tends to infinity. As was stated in the comments, whoever posted this most likely intended to include more information/restrictions.

Also, these two curves will not "meet at infinity." Both diverge as $x$ gets arbitrarily large


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