A rectangle prism is 7 cm long, 5 cm wide, and 4 cm. Which values are the areas of cross sections that are parallel to a face or
base of the prism
1 answer:
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First, it would be good to know that the area bounded by the curve and the x-axis is convergent to begin with.
Let
, so that 
, and the integral is equivalent to
The integrand is continuous everywhere except
, but that's okay because we have 
. This means the integral is convergent - great! (Moreover, there's a special function designed to handle this sort of integral, aptly named the "sine integral function".)
Now, to compute the volume. Via the disk method, we have a volume given by the integral
By the same substitution as before, we can write this as
The half-angle identity for sine allows us to rewrite as
and replacing
, 
, we have
Like the previous, this require a special function in order to express it in a closed form. You would find that its value is
where
is the Euler-Mascheroni constant and 
denotes the cosine integral function.
Let
The integrand is continuous everywhere except
Now, to compute the volume. Via the disk method, we have a volume given by the integral
By the same substitution as before, we can write this as
The half-angle identity for sine allows us to rewrite as
and replacing
Like the previous, this require a special function in order to express it in a closed form. You would find that its value is
where