It is worth revisiting the distinction between accumulation and net accumulation.
Figure 5.3.3 displays the graphs of $y=A(x)$ grouped by value of a. The first row shows $a=0$ with various values of $\Delta t$, the second row shows $a=2$ with various values of $\Delta t$, the third shows $a=1$, and the fourth row shows $a=3$.
Each row of Figure 5.3.3 illustrates the same tendency as we saw in Figure 5.3.2, that the displayed graphs of $y=A(x)$ became smoother and changed less as $\Delta t$ became smaller. This illustrates the second aspect of “close enough”. Letting $\Delta t=0.01$ is sufficient for us to have a clear idea of the behavior of
h, the exact displacement function that is generated from $r_h(x)=9.8x$.
Figure 5.3.3 illustrates another point that is discussed below. This is that changing the value of a not only changes the initial value at which net accumulation begins, changing the value of a also reveals more or less of the graph of the total accumulation function, taking into consideration that $A(a)=0$.
a) Define $r_f$ as $r_f(x)=x^2+0.5$. Vary the value of a. Explain the behavior of $y=A(x)$ in terms of the value of a and the values of $r_f$.
b) Define $r_f$ as $r_f(x)=x^23x+1$. Vary the value of a. Explain the behavior of $y=A(x)$ in terms of the value of a and the values of $r_f$.
c) Define $r_f$ as $r_f(x)=x(x1)(x2)$. Vary the value of a. Explain the behavior of $y=A(x)$ in terms of the value of a and the values of $r_f$.
d) Define $r_f$ as $r_f(x)=2xe^{\frac{x}{2}}$. Vary the value of a. Explain the behavior of $y=A(x)$ in terms of the value of a and the values of $r_f$.
 Let $r_f(x)=xe^{x}$.

Explain, in your own words, the meaning of “∫” in $\int_a^x r_f(t)dt$.

Explain, in your own words, the roles of x and t in $\int_a^x r_f(t)dt$.

Explain, in your own words, the meaning of $r_f(t)dt$ in $\int_a^x r_f(t)dt$.

Explain, in your own words, the meaning of $\int_a^x r_f(t)dt$.

Answer Exercise
5.2.4, part (d), using integral notation.

Density of a material at a point is defined as the rate of change of mass with respect to volume. The density of a concrete block at a point in its interior is 67 $\mathrm{\frac{g}{cm^3}}$. This means that if a small cube around that point were to increase in volume by 0.001 $\mathrm{cm}^3$, the mass within that volume would increase by (67)(0.001) kg.
 A concrete beam is painted on one end. It has uniform density in any cross section, but the density of the cross sections varies with distance from its painted end. The beam is 10m long and has a crosssection of 12cm x 5 cm. The rate of change function for the beam’s mass with respect to its distance from its painted end is $r_m(d) = 67 + 12\cos(3d)\sin(4d) \mathrm{\frac{g}{cm^3}}$ , 0≤d≤10 meters.

Define a function that gives the mass of the concrete beam as a function of distance from its painted end.
Graph your function in GC.

Click on a point on the graph. Record its coordinates. Interpret the coordinates in terms of the beam’s mass and in terms of $r_m$.

What is the beam’s mass that is made by the segment from 2 to 5 feet from its painted end?

We need to mark the place on the beam where 50% of its mass is on either side of the mark. Where should we place the mark?

(Adapted from Briggs & Cochran, Ch. 6.1, Problem 57).
 A Newton is the force required to accelerate a mass of 1 kg at the rate of 1 $\mathrm{\frac{m/sec}{sec}}$.
 Energy is commonly measured in units of Joules (J), which is the work done by a force of one Newton when its point of application moves one meter in the direction of the force.
 The rate at which energy is used is known as power. Power is measured in $ \mathrm {{J}/{sec}}$. 1 $\mathrm{\frac{J}{sec}}$ is called a Watt.
 It is also common to measure an amount of energy in kilowatthours (kWh). One kWh is the amount of energy used when it is used at the rate of 1000 $\mathrm{\frac{J}{sec}}$ (1000 Watts) for one hour.
 A megawatthour is the amount of energy used when it is used at the rate of one million Watts (one million $\mathrm{\frac{Joules}{sec}}$) for one hour. Or, more basically, a megawatthour is the amount of energy used to accelerate 1 million kg a distance of 1 m at the rate of 1 $\mathrm{\frac{m/sec}{sec}}$, for 1 hour.
A city consumes electrical energy on a given day at an approximate rate of $r_E(t)$, where $E(t)$ is the energy, in megawatthours, that the city consumes in t hours since midnight. The rate of change function $r_E$ is defined, in megawatts, as $$r_E(t) = 400  300\cos\left(\frac{\pi}{12}(t5)\right)\text{ if }0 \le t.$$
a) Fill in the blanks in
5
different ways to make each statement true:
3 Megawatthours = The energy consumed at a rate of __________ Joules/sec for __________ hours.
3 Megawatthours = The energy consumed at a rate of __________ Joules/sec for __________ hours.
3 Megawatthours = The energy consumed at a rate of __________ Joules/sec for __________ hours.
3 Megawatthours = The energy consumed at a rate of __________ Joules/sec for __________ hours.
3 Megawatthours = The energy consumed at a rate of __________ Joules/sec for __________ hours.
b) Why is it sensible that $\frac{\pi}{12}(t5)$ is the argument to cosine in this model of the city’s rate of electrical energy consumption? (Examine the graph of $r_E$.)
c) Define the function E that gives the electrical energy this city will have consumed x hours after midnight on a given day. Explain how your function produces a value that is in the desired units. Does the value of x have a necessary upper bound?
d) Approximately how much electrical energy does this city use in a typical day? In a typical week? In a typical month? In a typical year? Be sure to state quantity’s units.
e) This city’s electric utility company charges for electricity at the rate of $0.13 per kilowatthour
for electricity used between 7:00a and 5:00p, and at the rate of $0.07
per kilowatthour otherwise. What is this city’s electrical bill for
one day?
f) Burning 1 kg of coal produces about 450 kWh of energy. How many kg of coal are required to meet the energy needs of the city for one day? One year?
g) A wind turbine normally generates electricity at a rate of 200 kW. Approximately how many wind turbines would be required to meet the needs of this city for one day? One year?
h) A mediumsized household typically uses about 300500 kilowatthours of electrical energy in a month during the fall. Residential consumption is typically about 30% of a major city’s total electrical energy consumption in that same period. Approximately how large is this city’s population?

The discussion in Section 5.3.2 regarding the meaning of a in an approximate accumulation function applies to integrals, too. Explain why $$f(x) = f(a) + \int_a^x r_f(t)dt$$ for all values of x given any value of a in the domain of f.