= Solution
Write $X=\min(T,c)$ and $\Delta=\mathbf1_{\{T\le c\}}$ for the <right-censored> <survival time> and observed-failure indicator. Since the <exponential distribution> is continuous, including or excluding equality at $c$ makes no difference. Using the tail-integral expression for an <expectation>,
$$
E[X]=\int_0^cP(T>u)\,du=\int_0^c e^{-\theta u}\,du=\frac{1-e^{-\theta c}}{\theta}.
$$
Consequently the expected <integrated hazard> at the observed endpoint satisfies
$$
\boxed{E[H(X)]=\theta E[X]=1-e^{-\theta c}=P(T\le c)=E[\Delta].}
$$
<Administrative censoring> therefore stops both the accumulated exposure and the opportunity to record a failure.
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