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I am a freshman enrolled at an American University. Recently, I took an examination in which the following problem appeared:

Evaluate the following integral:

$int_0^4sqrt{16-x^2}dx$

My answer: 4$pi$, was correct. However, I received reduced credit for this answer because I did not solve it correctly (according to the professor). The exams are time-limited and have a fair amount of content, so when I saw this problem, 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$pi$.

The context of the test was surrounding our unit on inverse trigonometry and integration by parts. This section of the test did not list any other instructions besides evaluating the definite integrals. I've talked to my professor about it and his only response was that I solved it wrong:

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

Is my interpretation of evaluating the integral different? Does the instruction "Find the antiderivative and then evaluate" not need to exist for that to be required?

Thank you.

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.

As a student who has had a similar thing happen and heard of it happening to others, my personal recommendation would be to not bring it up again, it will probably be a losing battle. :(

But I would not consider your method "wrong" or "incorrect." There are many ways to solve a problem, you simply just made use of one of them, that wasn't the desired one.

As long as you explained how you came to your answer, the reason why your professor $textit{probably}$ marked you down is that based on the class and section the test covered, the question was designed so that you would display and make use of your knowledge of trig substitution to solve the problem.

I would hope your professor didn't take too many points off (since the method does work), but in these classes you will usually be expected to give the professor a specific method that they are looking for.

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.

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.