Day 2 — Calculus
Time: ~45 min · Topic: calculus (limits, derivatives, integrals)
The problem
Evaluate the integral from 0 to infinity of x^2 * e^(-x^2) dx
Hint: Try a substitution that simplifies x^2 into a single variable — the result should remind you of a very famous function from analysis.
Try it yourself
Take a few minutes. Don't peek at the steps. If you get stuck, the hint above is enough to nudge you forward.
Step-by-step solution
- Substitute u = x^2, so x = sqrt(u) and dx = du / (2*sqrt(u)). The integral becomes (1/2) * integral from 0 to infinity of u^(1/2) * e^(-u) du.
- Recognize this as (1/2) * Gamma(3/2), where the Gamma function is defined as Gamma(n) = integral from 0 to infinity of t^(n-1) * e^(-t) dt.
- Apply the Gamma function recursion: Gamma(3/2) = (1/2) * Gamma(1/2). The key identity Gamma(1/2) = sqrt(pi) gives us Gamma(3/2) = sqrt(pi)/2.
- Multiply it out: (1/2) * sqrt(pi)/2 = sqrt(pi)/4.
Answer: sqrt(pi)/4, which is approximately 0.4431.
Why this matters
This integral is a building block of the normal distribution — every time your phone predicts your commute time, it's secretly using a cousin of this exact calculation.
This day in math: The Mathematician Who Won Olympic Silver
April 22, 1887 · Harald Bohr
On this day in 1887, Harald Bohr was born in Copenhagen, Denmark. Before becoming one of the 20th century's most original mathematicians, Bohr was a star footballer who won a silver medal at the 1908 London Olympics. Playing for Denmark, he helped crush France 17-1 in the semi-final — still an Olympic record. When he later defended his doctoral thesis in mathematics, the audience reportedly contained more football fans than mathematicians.
Why it matters: Bohr went on to create the theory of almost periodic functions, a powerful generalization of periodic behavior that influences signal processing, differential equations, and number theory to this day. He was also the brother of Nobel Prize-winning physicist Niels Bohr — making them perhaps the most brilliant sibling duo in scientific history.
Did you know?
There's a shape called a Reuleaux triangle that has three curved sides and three pointy corners — yet it rolls as smoothly as a perfect circle. Even wilder: engineers use drill bits shaped like Reuleaux triangles to cut holes that are nearly perfect squares!
A Reuleaux triangle is constructed by drawing three circular arcs, each centered on one corner of an equilateral triangle and passing through the other two corners. This gives it a property called "constant width" — the distance between any two parallel lines squeezing the shape is always the same, just like a circle. When it rotates inside a square boundary, its corners trace a path that covers about 98.8% of the square's area, producing a nearly square hole.
Done?
- Solved the problem (or read through the steps)
- Read the history note
- Read the "did you know" fact