Recall that displacement \$d\$ is the area under the velocity curve. For a sinusoidal drift velocity \$v_d\$ having radian frequency \$\omega=2\pi f\$ where \$f=60\,\text{Hz}\$, the magnitude of maximum displacement over one half cycle can be calculated as the integral of \$v_d\$ with respect to time, during the time interval \$(0 \le t \le \pi/\omega)\,\text{s}\$:

$$
\begin{align*}
d &= \int_{0}^{\pi/\omega}v_d\,dt,\;\;v_d(t) = J(t) / (\rho_e\,e)\\
&= \frac{1}{\rho_e\,e}\int_{0}^{\pi/\omega}J(t)\,dt,\;\;J(t) = I(t)/A\\
&= \frac{1}{\rho_e\,e\,A}\int_{0}^{\pi/\omega}I(t)\,dt,\;\;I(t) = k\,\sin (\omega t)\\
&= \frac{k}{\rho_e\,e\,A}\int_{0}^{\pi/\omega}\sin(\omega t)\,dt\\
&= \frac{2\,k}{\rho_e\,e\,A\,\omega}
\end{align*}
$$

where \$k=0.1\,\text{A}\$ (as specified in the book example).

For what it's worth, when I crunch the numbers with MATLAB (see Listing 1 and Figure 1 below) the calculated displacement—i.e., drift distance—is approximately 12 nm; so I'm not sure how the authors arrived at the value 450 nm for the drift distance.

See also:

 - http://hyperphysics.phy-astr.gsu.edu/hbase/electric/ohmmic.html
 - https://activecalculus.org/single/sec-4-1-velocity-distance.html
 - https://pages.uncc.edu/phys2102/online-lectures/chapter-6-electric-current-and-resistance/6-3-drift-speed/

----------
Listing 1.  MATLAB source code

<!-- language: lang-matlab -->

    %% Housekeeping
    clc
    clear
    
    %% Givens
    d = 2.05e-3;            % wire diameter, m
    r = d/2;                % wire radius, m
    A = pi*(r^2);           % wire cross-sectional area, m^2
    
    q = 1.602e-19;          % electron charage, C
                            % (NB: This is 'e' in the equation above).

    n = 8.46e28;            % estimate of the number of charge-conducting 
                            % electrons per cubic meter in solid copper
                            % (NB: This is 'rho_e' in the equation above).
    
    k = 0.1;                % Sinusoidal current amplitude, peak
    f = 60;                 % Sinusoidal current frequency, Hz
    w = 2 * pi * f;         % Sinusoidal current frequency, rad/sec
    
    %% Equations
    % Current in the wire, C/s
    I = @(t)  k * sin(w*t);
    
    % Current density in the wire at time t, C s^-1 m^-2
    % J = I/A = k*sin(w*t)/A = k/A * sin(w*t)
    % Let k2 = k/A
    k2 = k/A;
    J = @(t)  k2 * sin(w*t);
    
    % Average electron drift velocity at time t, m/s
    % vd = J/n/q = I/n/q/A = k*sin(w*t)/n/q/A
    % Let k3 = k/n/q/A
    k3 = k/n/q/A;
    vd = @(t)  k3 * sin(w*t);
    
    % Average electron displacement at time t, m
    % displacement = k/n/q/A/w * (1 - cos(w*t))
    % Let k4 = k/n/q/A/w
    k4 = k/n/q/A/w;
    displacement = @(t)  k4 * (1 - cos(w*t));
    
    %% Solutions
    % For sin(w*t), max drift velocity occurs at w*t == pi/2 -> t = pi/2/w
    vd_max = vd( pi/2/w )
        % 2.2355e-06 -> ~2.2 um/s
    
    % Maximum average displacement of an electron during 1/2 cycle of 60 Hz 
    % can be calculated as the area under the drift velocity curve during 
    % the time interval (0 <= t <= pi/w) sec
    % NB: For sin(w*t), 1/2 cycle occurs at w*t == pi -> t = pi/w
    displacement_max = integral(vd, 0, pi/w )
        % 1.1860e-08 -> ~12 nm
    
    
    %% Plot the velocity and displacement curves vs time
    clf('reset')
    
    % NB: For sin(w*t), 1/2 cycle occurs at w*t == pi -> t = pi/w
    t_ = linspace( 0, pi/w );
    
    % drift velocity in micrometers/sec at time t
    vd_t = vd(t_) * 1e6;
    yyaxis left
    plot(t_, vd_t)
    
    % displacement in nanometers at time t
    displacement_t = displacement(t_) * 1e9;
    yyaxis right
    plot(t_, displacement_t)
    
    yyaxis left
    title('Velocity and Displacement vs time')
    xlabel('Time (sec)')
    ylabel('Velocity (um/s)')
    yyaxis right
    ylabel('Displacement (nm)')
    grid on

-----------

[![MATLAB plot of electron velocity and displacement vs. time][1]][1]

Figure 1.   MATLAB plot of electron velocity and displacement vs. time.


  [1]: https://i.sstatic.net/F5cEt.png