5128_Ch07_pp378-433

382 Chapter 7 Applications of Definite Integrals
What we learn from Examples 2 and 3 is this: Integrating velocity gives displacement
(net area between the velocity curve and the time axis). Integrating the absolute value of
velocity gives total distance traveled (total area between the velocity curve and the time
axis).
General Strategy
The idea of fragmenting net effects into finite sums of easily estimated small changes is
not new. We used it in Section 5.1 to estimate cardiac output, volume, and air pollution.
What is new is that we can now identify many of these sums as Riemann sums and express
their limits as integrals. The advantages of doing so are twofold. First, we can evaluate one
of these integrals to get an accurate result in less time than it takes to crank out even the
crudest estimate from a finite sum. Second, the integral itself becomes a formula that enables
us to solve similar problems without having to repeat the modeling step.
The strategy that we began in Section 5.1 and have continued here is the following:
Strategy for Modeling with Integrals
1. Approximate what you want to find as a Riemann sum of values of a continuous
function multiplied by interval lengths. If f x is the function and a, b the interval,
and you partition the interval into subintervals of length Δx, the approximating
sums will have the form f c k Δx with c k a point in the kth subinterval.
2. Write a definite integral, here b f x dx, to express the limit of these sums as
a
the norms of the partitions go to zero.
3. Evaluate the integral numerically or with an antiderivative.
EXAMPLE 4
Modeling the Effects of Acceleration
A car moving with initial velocity of 5 mph accelerates at the rate of at 2.4t
mph per second for 8 seconds.
(a) How fast is the car going when the 8 seconds are up?
(b) How far did the car travel during those 8 seconds?
SOLUTION
(a) We first model the effect of the acceleration on the car’s velocity.
Step 1: Approximate the net change in velocity as a Riemann sum. When acceleration
is constant,
velocity change acceleration time applied. rate of change time
To apply this formula, we partition 0, 8 into short subintervals of length Δt. On
each subinterval the acceleration is nearly constant, so if t k is any point in the kth
subinterval, the change in velocity imparted by the acceleration in the subinterval is
approximately
at k Δt mph. m ph
sec
sec
The net change in velocity for 0 t 8 is approximately
at k Δt mph.
Step 2: Write a definite integral. The limit of these sums as the norms of the partitions
go to zero is
8
at dt.
0
continued