= Solution
By <Fourier inversion>, the <sinc function> $\sin(At)/(\pi t)$ has <Fourier transform> equal to one for $|\omega|<A$ and zero for $|\omega|>A$. Thus the two bandwidths add to give
$$
\widetilde z(\omega)=\begin{cases}
2,&|\omega|<1,\\
1,&1<|\omega|<2,\\
0,&|\omega|>2.
\end{cases}
$$
The requested squared-magnitude <spectrum> therefore has heights four and one:
$$
\boxed{|\widetilde z(\omega)|^2=\begin{cases}
4,&|\omega|<1,\\
1,&1<|\omega|<2,\\
0,&|\omega|>2.
\end{cases}}
$$
Values at the jump points depend on the chosen Fourier convergence convention and do not affect any integral. The graph is shown below.
\Image[/past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2018/paper-1-q4-spectrum.png]
{title=Squared Fourier magnitude of the sum of two sinc signals}
The <Parseval identity> with this <Fourier transform> normalization gives
$$
\boxed{E=\frac1{2\pi}\int_{-\infty}^{\infty}|\widetilde z(\omega)|^2\,d\omega
=\frac1{2\pi}(4\cdot2+1\cdot2)=\frac5\pi}.
$$
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