Yes - Calculus can be taught visually, that's how my father taught it to me. I was a wiz at geometry, I can bisect lines and draw tangents in my mind.
Here's what I didn't understand though....what does the area under the curve have to do with anything? The line on graph paper was a line - what value was the area? To me the line was continuous - it didn't end, it was a function - so how could the area have bounds?
When I was given min-max problems in College the area/vol was always something concrete (e.g. land size, a rectangle, or a water bottle). I had a difficult time with Calc in college because I just couldn't relate these "areas under the curve" to anything real. I could do the mechanics (integrate, derivatives etc) and understood acceleration/speed. It wasn't until I was older that some of these area/volumes started to make sense (What is "work?" :-D )
My suggestion - I can't be alone in this problem - is to relate these areas to things. Answer the question: why is the area equal/equivalent/describe X ?. I had to take it on faith - my Dad said so. Can this be shown or described and be shown to "be really the answer" -- Why is it that?! A bit more concrete evidence that this is true.
I may not be an abstract thinker in math. This is why I program computers ;-)