Reddit reviews Calculus, 5th Edition
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CalculusJames Stewart 5th EditionInteractive Video Skillbuilder CD for Stewart's Calculus 5th EditionReference pages in the front and back of the bookThomson/Brooks/Cole
If you are quite comfortable with the various techniques of integration, I don't think you should have too much trouble, as I recall that's the basic focus of calc 2, or at least it was when I went through it (ca. 2001). I wouldn't recommend skipping it if you're shaky on it, as getting behind can be quite hard to recover from, but I don't know where you are. To be fair calc 3 doesn't build directly on calc 2 as much as 1, but if you're going into engineering I imagine they'll be quite a boon to have down solidly.
You might consider asking your advisor to connect you with some faculty who can give you a reasonable idea what's expected. The Math department faculty are by and large quite realistic and down-to-earth people and should be quite up front with you about whether it'd work for you.
At the time (and this may have changed) they were using some version of Stewart Calculus so if you can find a copy of that somewhere, you can go through some problems and see if you're comfortable with the material that class covers. Any faculty teaching the class can tell you what chapters are covered and in which edition.
Best of luck and welcome to Cal Poly!