232
4.2. RIEMANN SUMS
(b) How does your answer to (a) change if S is a left Riemann sum? a middle
Riemann sum?
(c) Suppose that S really is a right Riemann sum. What is geometric quantity does
S approximate?
(d) Use sigma notation to write a new sum R that is the right Riemann sum for the
same function, but that uses twice as many subintervals as S.
3. A car traveling along a straight road is braking and its velocity is measured at several
different points in time, as given in the following table.
seconds, t
0
0.3 0.6 0.9 1.2 1.5 1.8
Velocity in ft/sec, v(t) 100 88 74 59 40 19 0
(a) Plot the given data on a set of axes with time on the horizontal axis and the
velocity on the vertical axis.
(b) Estimate the total distance traveled during the car the time brakes using a
middle Riemann sum with 3 subintervals.
(c) Estimate the total distance traveled on [0, 1.8] by computing L 6 , R 6 , and
1
2 (L 6 + R 6 ).
(d) Assuming that v(t) is always decreasing on [0, 1.8], what is the maximum
possible distance the car traveled before it stopped? Why?
4. The rate at which pollution escapes a scrubbing process at a manufacturing plant
increases over time as filters and other technologies become less effective. For this
particular example, assume that the rate of pollution (in tons per week) is given by the
function r that is pictured in Figure 4.18.
1
2
3
4
1
2
3
4
y = r(t)
weeks
tons/week tons/week
Figure 4.18: The rate, r(t), of pollution in tons per week.
(a) Use the graph to estimate the value of M 4 on the interval [0, 4].
4.2. RIEMANN SUMS
(b) How does your answer to (a) change if S is a left Riemann sum? a middle
Riemann sum?
(c) Suppose that S really is a right Riemann sum. What is geometric quantity does
S approximate?
(d) Use sigma notation to write a new sum R that is the right Riemann sum for the
same function, but that uses twice as many subintervals as S.
3. A car traveling along a straight road is braking and its velocity is measured at several
different points in time, as given in the following table.
seconds, t
0
0.3 0.6 0.9 1.2 1.5 1.8
Velocity in ft/sec, v(t) 100 88 74 59 40 19 0
(a) Plot the given data on a set of axes with time on the horizontal axis and the
velocity on the vertical axis.
(b) Estimate the total distance traveled during the car the time brakes using a
middle Riemann sum with 3 subintervals.
(c) Estimate the total distance traveled on [0, 1.8] by computing L 6 , R 6 , and
1
2 (L 6 + R 6 ).
(d) Assuming that v(t) is always decreasing on [0, 1.8], what is the maximum
possible distance the car traveled before it stopped? Why?
4. The rate at which pollution escapes a scrubbing process at a manufacturing plant
increases over time as filters and other technologies become less effective. For this
particular example, assume that the rate of pollution (in tons per week) is given by the
function r that is pictured in Figure 4.18.
1
2
3
4
1
2
3
4
y = r(t)
weeks
tons/week tons/week
Figure 4.18: The rate, r(t), of pollution in tons per week.
(a) Use the graph to estimate the value of M 4 on the interval [0, 4].