I would say that while math is quite helpful, "being a whiz" is not a strict requirement for subfields like software engineering. Being a problem-solver--tenacious, curious, willing to dig and poke and try different approaches to understand a problem and how to deal with it--absolutely is, however, and things like statistics, boolean logic, and complexity analysis/"big-O notation" (all tiny subsets of math) definitely play a part.
On the other hand, I'm pretty sure there's a strong correlation between those who find the logic of mathematics clear and those who do well in engineering. Even if engineering success doesn't necessarily require a clear understanding of math, those who lack it are probably going to have to work a lot harder to be successful at engineering.
Finally, the more physically oriented forms of engineering (aeronautical, mechanical, civil) do have a much stronger dependency on math. Software engineering may actually be the biggest outlier; bioengineering may be another. (I was a physicist, but I lived with a lot of engineers in college. I don't know as much about present-day bioengineering.)
Christina J. Bryz-Gornia, PE
I agree with Greg's comment regarding the correlation between math and logical thinking. I don't think you need to be a "whiz," but you do need to have a strong ability to draw conclusions from a problem. Research, critical thinking, and inference of solutions are absolutely essential skills. It's amazing how much engineering requires thinking critically, logically, and working through (the many steps of) a problem. Yes math is important, but critical thinking is arguably more valuable.
In terms of completing an undergraduate degree in engineering, the programs I've seen as an academic and transfer counselor require quite a bit of math. I was a Liberal Arts major myself, so this sounds like high-level math to me. But that will obviously depend on your general comfort level with the subject.
Students at the community college where I work may be required to complete Calculus with Analytic Geometry I and II (10 credits worth of math). If they have to take prerequisites to get there, that might mean another five or more credits (Precalculus with Trigonometry), depending on how they place on the math placement tests. After the Calculus with Analytic Geometry (and depending on your transfer school of choice), it might also mean Differential Equations, Linear Algebra, etc.
I'd say it's definitely one of the more math-intensive majors. So the question really becomes whether you feel strongly enough about the actual work of BEING an engineer to get through those math courses. The other thing to consider, of course, is that those classes are required because, in at least some cases, they reflect the skill sets you'll actually use on the job. I imagine actual engineers may say you really don't. But I also imagine that they still use a lot more math in their day-to-day work than I do as an academic counselor, for instance. It's all relative.
Short answer: You'll take quite a few math courses on your way to earning a degree in engineering.