Calc 3 for computer science?

I'm going to be a freshman studying computer science at Madison this fall, and I'm stuck trying to decide which math course to take. I completed calc 1 and 2 in highschool, and I had originally assumed calc 3 would be a required course, but I've since learned that not only is it not required, the content taught is mostly unapplicable to the computer science major. This made me think it would be a better idea to take linear algebra (340) my first semester instead of calc 3 to spearhead myself into the practical side of math for computer science.

I'm writing this to ask whether not taking calc 3 would block my ability to learn higher level cs courses in the future, like graphics, computer vision, and machine learning, OR if I could self study whatever math I need for those classes which was ONLY covered in calc 3. I don't really want to take a gap between calc 2 and 3, because that would only make the class more difficult for me.

It's also important to note that currently, my goal is to graduate in 3 years (not a problem with previous credits) and go into the industry. I am interested in doing research while I'm at school, but I DON'T plan on getting a PHD or going into academia after college. Even so, I would still like to understand these topics in case I ever want to work with them in the corporate world in the future.

On a side note, how worth it would a double major in math (on the Computational track) be considering my goals? I know calc 3 definitely is a requirement in that case.

Thanks in advance for any advice!