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 + ... + (b-1)^2 + b^2

Now, we know that

n

∑ k^2 = n(n+1)(2n+1)/6

k=1

So, we want sum from a-b = sum from 1-b - sum from 1-a, or

b(b+1)(2b+1)/6 - a(a+1)(2a+1)/6

= 1/6 (2b^3+3b^2+b - 2a^3-3a^2-a)

= 1/3 (b^3-a^3) plus some lower-order junk

So, our integral exact value appears to agree with the integer approximation.

## Answer this Question

## Related Questions

- Algebra 2 - Can you double check my answer The lengths of the sides of a ...
- math squares - how many squares with sides that are 6 inches long are needed to ...
- Calculus - A retangular poster is to be made out of cardboard. The margins on ...
- Math - 1) Rectangle ABCD has a perimeter of 106 inches. If the width is 7 inches...
- Pre-Calculus - A window has the shape of a square surmounted by an isosceles ...
- Math - A roll of paper towels is 10 inches high and 2.5 inches in diameter. How ...
- One Trigonometry Question Please Help! - Quilt squares are cut on the diagonal ...
- Math ASAP - Height: 10 inches Base Perimeter: 40 inches Lateral Surface Area: ...
- Math-Algebra 1 - I Really Need Help With This Question!! What is the area of the...
- Math-Algebra 1 - I Really Need Help With This Question!! What is the area of the...