1. Describe how you could approximately find the total number of particles in a rectangular box that is 2m x 2m x 3m by making no more than 12 samples.

Question

2. You know the following facts about a function f(x): The derivative of f(x) is 6x + 4. The integral of f(x) from 1 to 3 is 40. Find f(5).

Answer 1

1. I have no clue what you mean by "particle" or what the question asks for.

#2 is straighforward

since f'(x) = 6x+4
f(x) = 3x^2 + 4x + c

⌠(3x^2 + 4x + c)dx from 1 to 3 = 40
⌡

= x^3 + 2x^2 + cx│ from 1 to 3 = 40

so

(27+18+3c)-(1+2+c) = 40
.
.
c=-1

then f(x)=3x^2 + 4x - 1
and f(5) = 94