Area Under the Curve
Comprehension — Area under normal curve
Grade 12
Question:
<p>If \(\int_0^1 e^{-x^2/2}\,dx\approx0.3413\cdot\sqrt{2\pi}\) (one standard deviation), which best describes \(\int_0^\infty e^{-x^2/2}\,dx\)? [MAU042]</p>
\sqrt{\pi/2}
\pi/2
\sqrt\pi
1
Step-by-Step Solution
Key Concept: \int_0^\infty e^(-x^2/2)dx: let t=x/\sqrt{2}, dt=dx/\sqrt{2.} = \sqrt{2} \cdot \int_0^\infty e^(-t^2)dt = \sqrt{2} \cdot (\sqrt\pi/2) = \sqrt{\pi/2}.
<div class='solution'>
<p>Let $t=x/\sqrt{2}$, $x=\sqrt{2}t$, $dx=\sqrt{2}\,dt$:</p>
<p>$$\int_0^\infty e^{-x^2/2}dx=\sqrt{2}\int_0^\infty e^{-t^2}dt=\sqrt{2}\cdot\frac{\sqrt{\pi}}{2}=\sqrt{\frac{\pi}{2}}$$</p>
</div>
Correct Answer: ['A']