system
A chat on subreddit /r/me_irl
human
Ok, so the first thing you ask yourself when working on a math problem is "What am I solving for?" Here, we are solving for integrals. First, distribute the terms in parentheses, and then focus on the integral of x3 cos (x/2) √(4 – x2). This entire integral is equal to 0.



x3 is an odd function
cos (x/2) is an even function
√(4 – x2) is an even function

The product of an odd function and an even function is an odd function, so the product of these three functions is an odd function. The integral of an odd function over a symmetric interval [-a, a] is equal to 0. Hence, this integral is equal to 0, makes sense right?

Now this is where we focus on the integral of (1/2) √(4 – x2) from -2 to 2.



The integral of √(4 – x2) from -2 to 2 is the area of a semi-circle, and we want half of that. So this means that the integral evaluates to:

(1/2)(1/2)π(2)2 = π

So the entire integral evaluates to π. But here comes another question, how can we remember the first 8 digits of pi?

One neat trick my high school teacher taught me is to remember a sentence where the number of letters in each word gives the digits of pi. For example:

How I wish I could calculate pi easily today

The number of letters in each word is

3 How

1 I

4 wish

1 I

5 could

9 calculate

2 pi

6 easily

5 today

The password is the first 8 digits, which is 31415926.

Hope this helps
gpt
r/theydidthemath