Posted by Anon on Sunday, March 16, 2014 at 6:14pm.
The average area of all squares with sides between a inches and b inches (b>a) is ____ in^2.

AP Calculus  Steve, Sunday, March 16, 2014 at 11:46pm
∫[a,b] x^2 dx
= 1/3 x^3 [a,b]
= 1/3 (b^3  a^3)
Consider only integer values. Then that will give you
a^2 + (a+1)^2 + ... + (b1)^2 + b^2
Now, we know that
n
∑ k^2 = n(n+1)(2n+1)/6
k=1
So, we want sum from ab = sum from 1b  sum from 1a, or
b(b+1)(2b+1)/6  a(a+1)(2a+1)/6
= 1/6 (2b^3+3b^2+b  2a^33a^2a)
= 1/3 (b^3a^3) plus some lowerorder junk
So, our integral exact value appears to agree with the integer approximation.