I'm going to assume you have a TI84 Plus CE calculator.
This is a fairly complicated function and I think it makes sense to break it into pieces in order to make sure you get it right.
Let's let Y1 = the coefficient 1/(σ√(2π)) so, since you are given that σ = 117, enter this into your Y1:
Y1 = 1/(117√(2π))
Then let's let Y2 = the exponent -0.5((x-μ)/σ)2 and since you are given μ = 528 and σ = 117 then enter this into Y2:
Y2 = -.5((x-528)/117)2
Then let's make Y3 the total equation so enter:
Y3 = Y1•eY2
I set my window as follows:
Ymax = 0.004
To answer the questions, you can use the integration button: "math", then "9: fnInt("
For problem i) do
(Note that you get to the Y3 button by hitting the "vars" button, then scrolling right to "Y-VARS", then "1: Function", then scrolling down to "3:Y3"
and you will get 0.5
For problem ii), do the same except integrate from 0 to 800. You will get 0.98996
For iii) I actually am not sure whether they mean between the mean and 1 std dev or plus/minus 1 std dev from the mean. If you assume it is plus/minus 1 std dev, then integrate between -1σ to +1σ which is (528-117) to (528+117) or 411 to 645. If you want the mean to 1σ, then integrate between 528 and 645. Then repeat for 2σ.
For iv) integrate between 200 and 600
For v) you are looking for the integral from 0 to some number that equals 0.9000 so guess the upper limit of integration until you get 0.9000 as an answer.(Hint: try 678)