Scan through the array to build an out-of-order heap, that is, the first element is indeed the smallest, but necessarily any of the other elements. This takes us O(n). Then extract from the heap k times to get the smallest k elements using O(klogn). Thus the total time is O(n+klogn).
Other solution: Create a heap using bubble down per 4.3.4 in the book. This takes O(n) time. Then, extract from the heap in k times to get the smallest k elements. This takes O(k*logn) time. Therefore, total time is O(n + k*logn)