Alan

"A cubic polynomial p(x) has leading coefficient 1, all real coefficients, and p(3 - 2i) = 0. If p(0) = −52, find p(x)."

Can you take it from here?

This constraints represent a sphere of radius 1 inside a cube of side length 2.

Volume of cube  \(Vc=2^3=8\)

Volume of sphere \(Vs=\frac{4}{3}\pi 1^3=\frac{4}{3}\pi\)

Probability \(p=\frac{Vs}{Vc}\)

I'll let you finish!

Your result is correct.  I was too hasty!

Here's the approach for part A

Your integration for part B is correct.  (edited to correct my mistake!).

Presumably this should read

"There are 24 red beads. The number of red beads is 34 less than the number of blue beads. The number of white beads is 0.5 the number of blue beads. What is the total number of beads on Kirsten’s necklace?"

If so, then

r = 24

b = r + 34

w = b/2

total = r + b + w

Can you take it from here?

As follows:

Get to know the properties of logarithms!

As follows:

Like so:

Apologies, I misread your question! 

Unfortunately, there is no general rule for arbitrary functions.  For some functions you can find an expression for the sum, for others you can't.  Do you have some specific functions in mind?

log(b/a) = log(b) - log(a)

so

log₂(2/1) = log₂(2) - log₂(1)

log₂(3/2)=log₂(3) - log₂(2)

...

log₂(2010/2009) = log₂(2010) - log₂(2009)

Adding up all these, resuts in most of them cancelling out, and we are left with

log₂(2010) - log₂(1)  

Now 2¹⁰ = 1024 and 2¹¹ = 2048, so I'll leave you to decide the final steps.