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.

TIJTORIAL 11 Theorem (Monotonicity). Let R be a rectangle in Rn. Let f:RR and g:RR be integrable on R in Rn. If fg on R, then RfdVRgdV. 1. Below is a WRONG proof of monotonicity. 1. Let P be a partition of the rectangle R. 2. Since f is integrable on R,RfdV exists and satisfies RfdVUP(f). 3. Since g is integrable on R,RgdV exists and satisfies LP(g)RgdV. 4. Since fg on R, it follows that UP(f)LP(g). 5. Therefore, RfdVUP(f)LP(g)RgdV 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