  We consider a $\frac{1}{2}$-BPS solution for a D3 brane probe in AdS$_5
\times S^5$ that has world-volume geometry of AdS$_3 \times S^1$. It intersects
the boundary over a surface that represents a dimension 2 defect in the
boundary N=4 SYM theory. The effective action of the probe brane is
proportional to the logarithmically divergent volume of AdS$_3$ and may thus be
interpreted as computing conformal anomaly of the supersymmetric $S^2$ defect.
The classical action scales as $N$. We compute the 1-loop correction to it due
to quantum fluctuations of the D3 brane world-volume fields and compare the
result to an earlier suggested expression for the defect anomaly. We also
perform a similar analysis of a $\frac{1}{2}$-BPS M5 brane probe solution in
AdS$_7 \times S^4$ with the world-volume geometry of AdS$_5 \times S^1$ that
represents a dimension 4 defect in the boundary (2,0) 6d theory. Here the
classical M5 brane action computes the leading order $N^2$ term in $a$-anomaly
of the supersymmetric $S^4$ defect. We perform a detailed computation of the
1-loop correction to the M5 brane effective action and thus provide a
prediction for the subleading constant in the $S^4$ defect $a$-anomaly
coefficient.
