Calculus II Professor will not accept my correct integral evaluation that uses a different method, should I bring this up further?

I've done some undergraduate teaching and my policy is always if you get the correct answer by any means then you get full credit, but others have different policies and it's really up to them.

You could argue your case. Your professor could argue back that solving the integral by trig substitution does not require the formula $A = \pi r^2$, and he did not permit the use of that formula. He could argue that using that formula entails circular reasoning (the formula for the area of a circle has to be gotten by some limiting or integration method equivalent to evaluating $\int_{-r}^r \sqrt{r^2-x^2}dx$).

It could go either way for you. But I think it would be a waste of your and your professor's time.

An argument could be made that you should include a proof that the integral evaluates the area of a half-disk, rather than just asserting the answer.

Whether you “should” have gotten full points is more a matter of pedagogy than of mathematics, but as a practical tip: using (correct) method Y to solve a problem with instructions to use method X (especially in an intro class and when you are not familiar with the instructor and their teaching philosophy) is always a gamble.

I just wanted to add to the chorus of voices proclaiming that you're right and that your professor is completely wrong.

The evaluation of an integral should be taken as valid, as long as valid mathematical techniques are employed and shown to be employed. This holds as long as the question did not explicitly stipulate or forbid certain method(s). Just haing the test on a topic titled "Trigonometric integrals" is not sufficient. I'll expand on this point in a bit.

To see why this professor's "logic" leads to a slippery slope, consider this question:

"Evaluate the integral $\displaystyle \int_0^{\frac{\pi}{2}} \sin^2 x dx$".

Perhaps most students might evaluate this in the most obvious way, using a double angle cosine identity and plodding through.

But suppose a bright student does this:

"Let $I = \displaystyle \int_0^{\frac{\pi}{2}} \sin^2 x dx$.

By substituting $y = \frac{\pi}{2} - x$, we can see that $I = \displaystyle \int_0^{\frac{\pi}{2}} \cos^2 y dy = \displaystyle \int_0^{\frac{\pi}{2}} \cos^2 x dx$ since the variable of integration in a single variable definite integral is a dummy variable.

Hence $2I = \displaystyle \int_0^{\frac{\pi}{2}} \sin^2 x + \cos^2 x dx = \int_0^{\frac{\pi}{2}} 1dx = \frac{\pi}{2}$.

Therefore $I = \frac{\pi}{4}$."

Now this is a perfectly valid and highly elegant solution. It even uses "calculus-ey" working rather than a graphical shortcut. But something tells me your professor may not be the sort to accept this sort of creative solution, either, and his reasons for the rejection may be equally arbitrary.

We should never aim to discourage creativity in mathematics, only temper it with rigour.

Now, getting back to your situation, there are ways the professor could've avoided any ambiguity about how he wanted the question solved. For example:

"By using an appropriate trigonometric substitution, evaluate $\displaystyle \int_0^4 \sqrt{16-x^2}dx$",

in which case I would be on the side of your professor if you'd use the circular area to work out your integral.

Finally, a word of encouragement. As a fairly good student of math and physics, I've encountered rather poor behaviour from educators. I've had a math teacher marking my perfectly valid working wrong because of her own lack of knowledge. I've had a physics instructor ordering me to show my solution to an Olympiad prep question on the blackboard in front of the class, derisively refusing to accept my insistence that the question itself was incorrect, sending me back to my seat in embarrassment, and then later circulating a memo to the students in the class confirming exactly what I'd said (that the question was wrong), but not even acknowledging my existence, let alone rendering an apology for his conduct.

I've had some great teachers too, so it's not that I have a chip on my shoulders about teachers in general or anything. But we must acknowledge that teachers are human, with human foibles. And if we were to become teachers ourselves, we must try not to make the mistakes we've observed in others.

I noticed it was the equation of the top half of a circle centered at (0, 0) and with radius 4. Knowing this, and my knowledge of the integral indicating the signed area under a curve, I merely took the area of a quarter-circle of radius 4, $\frac{1}{4}$$\pi$$r^2$ and wrote my answer of 4$\pi$.

Did you write this clearly in your test (as you did here)? If not, it is fair to give reduced points. One should always explain where answers come from. If yes, proceed reading.

To receive full credit, you would have had to evaluate an integral, as the instructions indicated.

Do the instructions clearly disallow your solution? If not, it was not fair to you, and you should insist on it. If yes, read further.

Were these instructions available a priori, or they were included in the test itself? If available a priori, you should have complained about them before the test. If not, the instructions are unfair, and you should try to insist about it as well.

Also, recall that most of this is up to the professor, so you might be with bad luck, sadly.