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The standard deviation for the number of times an odd number is rolled is 15.8
<h3>How to determine the standard deviation?</h3>The given parameters are:
Die = regular six-sided die
n = 1000
The probability of rolling an odd number is:
p = 1/2 = 0.5
The standard deviation is then calculated as;
This gives
Evaluate the products
Evaluate the root
Hence, the standard deviation is 15.8
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I've attached a plot of one such cross-section (orange) over the region in the x-y plane (blue), including the bounding curves (red). (I've set 
for this example.)
The length of each cross section (the side lying in the base) has length determined by the horizontal distance
between the y-axis 
and the curve 
. In terms of 
, this distance is 
. The height of each cross section is twice the value of , so the area of each rectangular cross section should be 
.
This means the volume would be given by the integral
The length of each cross section (the side lying in the base) has length determined by the horizontal distance
This means the volume would be given by the integral