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TIJTORIAL Theorem (Monotonicity). Let R be a rectangle in R^(n) . Let f:R->R and g:R->R be integrable on R in R^(n) . If f on
TIJTORIAL\ Theorem (Monotonicity). Let
R
be a rectangle in
R^(n)
. Let
f:R->R
and
g:R->R
be integrable on
R
in
R^(n)
. If
f on
R
, then
\\\\int_R fdV.\ Below is a WRONG proof of monotonicity.\ Let
P
be a partition of the rectangle
R
.\ Since
f
is integrable on
R,\\\\int_R fdV
exists and satisfies
\\\\int_R fdV.\ Since
g
is integrable on
R,\\\\int_R gdV
exists and satisfies
L_(P)(g).\ Since
f on
R
, it follows that
U_(P)(f).\ Therefore,
\\\\int_R fdV as required.\ Identify the flawed line and explain why it is wrong.\ 2. Write a correct proof. Hint: Switch the roles of upper and lower. Pick two partitions. Use a common refinement.
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