5128_Ch07_pp378-433

Section 7.1 Integral as Net Change 385
Hooke’s Law for springs says that the force it takes to stretch or compress a spring x
units from its natural (unstressed) length is a constant times x. In symbols,
F kx,
where k, measured in force units per unit length, is a characteristic of the spring called the
force constant.
EXAMPLE 7 A Bit of Work
It takes a force of 10 N to stretch a spring 2 m beyond its natural length. How much work
is done in stretching the spring 4 m from its natural length?
SOLUTION
We let Fx represent the force in newtons required to stretch the spring x meters from
its natural length. By Hooke’s Law, Fx kx for some constant k. We are told that
The force required to stretch the
F2 10 k • 2,
spring 2 m is 10 newtons.
so k 5 Nm and Fx 5x for this particular spring.
We construct an integral for the work done in applying F over the interval from x 0
to x 4.
Step 1: Riemann sum. We partition the interval into subintervals on each of which F is
so nearly constant that we can apply the constant-force formula for work. If x k
is any point in the kth subinterval, the value of F throughout the interval is
approximately Fx k 5x k . The work done by F across the interval is
approximately 5x k Δx, where Δx is the length of the interval. The sum
Numerically, work is the area under the
force graph.
Fx k Δx 5x k Δx
approximates the work done by F from x 0 to x 4.
Steps 2 and 3: Integrate. The limit of these sums as the norms of the partitions go to
zero is
4
Fx dx 4
5xdx 5 x 4
2
40 N • m.
2
0
0
]
0
Now try Exercise 29.
We will revisit work in Section 7.5.
Quick Review 7.1 (For help, go to Section 1.2.)
In Exercises 1–10, find all values of x (if any) at which the function
changes sign on the given interval. Sketch a number line graph of the
interval, and indicate the sign of the function on each subinterval.
Example: f x x 2 1 on 2, 3
f (x)
+ – +
–2 –1 1 3
Changes sign at x 1.
1. sin 2x on 3, 2 Changes sign at p 2 ,0, p 2
2. x 2 3x 2 on 2, 4 Changes sign at 1, 2
x
3. x 2 2x 3 on 4, 2 Always positive
4. 2x 3 3x 2 1 on 2, 2 Changes sign at 1 2
5. x cos 2x on 0, 4 Changes sign at p 4 , 3 p 5p ,
4 4
6. xe x on 0, ∞ Always positive
x
7. x 2 on
1
5, 30 Changes sign at 0
8. x 2
2
x2
on
4
3, 3 Changes sign at 2, 2, 2, 2
9. sec 1 1 sin x 2 on ∞, ∞
1
10. sin 1x on 0.1, 0.2 Changes sign at 3 p , 1
2p
9. Changes sign at 0.9633 kp, 2.1783 kp, where k is an integer