Unraveling My Teacher's Math Questions: Explaining the Answers

[ƒ(x)]

These questions are ones that my teacher never explained:

1) if c is a positive integer, then the limit of (1/c)(1/c + 2/c ... + 5c/c) as c approaches infinity can be expressed as

a) integral of x^2 dx from 0 to 1
b) integral of 1/x dx from 0 to 5
c) integral of x dx from -5 to 5
d) integral of x dx from 0 to infinity
e) integral of x dx from 0 to 5

2) the limit as n approaches infinity of (1/n)( (n/1)² + (n/2)² + ... + (n/n)²) =

a) the integral of 1/x² dx from 0 to 1
b) the integral of 1/x dx from 0 to 1
c) the integral of x dx from 0 to 1
d) the integral of x² dx from 0 to 1
e) none of the above

3) if u is a positive integer, then the limit of (1/u)( (2/u)² + (4/u)² + ... + (8u/u)²) can be expressed as

a) the integral of 8/x² dx from 0 to 1
b) the integral of 1/x² dx from 0 to 1
c) the integral of 1/x² dx from 0 to 8
d) the integral of x²/2 dx from 0 to 8

I later found out that the answers were e,a, and d in that order but still need an explanation.

 

[ƒ(x)]

Ok. Now I realize where the start and end points for the integral come from. But not the rest.

 

[ƒ(x)]

Anyone?

Also, I think its the limit of the sum of ... whatever expression is there.