Is this another case of your teacher insisting that you use a more complicated method than necessary?
If the graph were bounded on the y-axis from, say, 0 to 3, I'd agree with you. But the graph is bounded from 0 to 4. Plug in any value you want for x and you'll see that the y-value never goes above 4. The boundaries set by the problem are meaningless because the graph never goes beyond those boundaries anyway.
Oh, and by the way, your integration is wrong. It's the correct integration for 16 - (x squared), but you forgot that the entire equation is under a square root.
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