= Solution
{c}
Take the inner surface to be free of <traction>, as appropriate to an empty cavity with no <surface tension>. At the outer surface the imposed <pressure> is compressive when positive. The outward <normal vectors> are $-\mathbf e_r$ at $r=a$ and $+\mathbf e_r$ at $r=b$, giving
$$
t_r(a)=0,\qquad t_r(b)=-P.
$$
Integrating the <ordinary differential equation> from part (ii) between the two surfaces gives
$$
-P=\int_{\lambda_a}^{\lambda_b}\frac{h'(\lambda)}{1-\lambda^3}\,d\lambda.
$$
The outer stretch follows from <incompressibility>:
$$
\lambda_b^3=\frac{b^3}{B^3}
=1+\frac{A^3}{B^3}(\lambda_a^3-1).
$$
Hence an expression entirely in terms of the inner stretch and the reference radii is
$$
\boxed{P(\lambda_a)=
\int_{\lambda_a}^{[1+(A/B)^3(\lambda_a^3-1)]^{1/3}}
\frac{h'(\lambda)}{\lambda^3-1}\,d\lambda.}
$$
Equivalently $P=-\int_{\lambda_b}^{\lambda_a}h'(\lambda)/(\lambda^3-1)\,d\lambda$. For an expanding shell, $1<\lambda_b<\lambda_a$; when $h'>0$ on this range, $P<0$, correctly describing an outward tensile loading. If an inner <pressure> $P_i$ were present, the same derivation would give $P-P_i$ on the left. Thus the empty-cavity condition specifies the pressure reference needed for the requested answer.
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