A nonintegrable function Consider the function defined on [0, 1] such that f ( x ) − 1 if x is a rational number and f ( x ) = 0 if x is irrational. This function has an infinite number of discontinuities, and the integral ∫ 0 1 f ( x ) d x does not exist. Show that the right, left, and midpoint Riemann sums on regular partitions with n subintervals equal 1 for all n . ( Hint: Between any two real numbers lie a rational and an irrational number.)
A nonintegrable function Consider the function defined on [0, 1] such that f ( x ) − 1 if x is a rational number and f ( x ) = 0 if x is irrational. This function has an infinite number of discontinuities, and the integral ∫ 0 1 f ( x ) d x does not exist. Show that the right, left, and midpoint Riemann sums on regular partitions with n subintervals equal 1 for all n . ( Hint: Between any two real numbers lie a rational and an irrational number.)
Solution Summary: The author shows the right, left and midpoint Riemann sums on regular partitions with n subintervals equal 1 for all.
A nonintegrable function Consider the function defined on [0, 1] such that f(x) − 1 if x is a rational number and f(x) = 0 if x is irrational. This function has an infinite number of discontinuities, and the integral
∫
0
1
f
(
x
)
d
x
does not exist. Show that the right, left, and midpoint Riemann sums on regular partitions with n subintervals equal 1 for all n. (Hint: Between any two real numbers lie a rational and an irrational number.)
Problem 11 (a) A tank is discharging water through an orifice at a depth of T
meter below the surface of the water whose area is A m². The
following are the values of a for the corresponding values of A:
A 1.257 1.390
x 1.50 1.65
1.520 1.650 1.809 1.962 2.123 2.295 2.462|2.650
1.80 1.95 2.10 2.25 2.40 2.55 2.70
2.85
Using the formula
-3.0
(0.018)T =
dx.
calculate T, the time in seconds for the level of the water to drop
from 3.0 m to 1.5 m above the orifice.
(b) The velocity of a train which starts from rest is given by the fol-
lowing table, the time being reckoned in minutes from the start
and the speed in km/hour:
| † (minutes) |2|4 6 8 10 12
14 16 18 20
v (km/hr) 16 28.8 40 46.4 51.2 32.0 17.6 8 3.2 0
Estimate approximately the total distance ran in 20 minutes.
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