# Area between parabola and a line that don't intersect? 0 or infinity

Mathematics Asked on January 5, 2022

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.

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

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

Answered by N. Bar on January 5, 2022

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