Volume of a semi-circle function plus constant

Question

So I have this function

Now when I try to rotate it around the x-axis between the intervals [148,160] 360 degrees, the volume contains imaginary answers i.e.

Now since we are talking about volume how can it be that when you rotate a function like mine around the x-axis 360 degrees, that it turns out to have imaginary constants?

Did I do anything wrong? If so how would I even begin to evaluate the volume of the function around this interval?

Answer 1

The reason is that when $x> 150 + \sqrt{50} $, the value under the square root is negative, hence is complex valued.

Thus, if you're looking at the "rotation of the real-valued graph", you want to restrict to the range $[148, 150 + \sqrt{50} ]$.

---

Note: It is reasonable to ask what is the volume of rotation of the line $y = i$ about the x-axis. So, check for what the problem is asking about.