First we have acceleration of a rocket with changing mass. In Physics HL we only ever deal with changing accelerations by looking at graphs but I wanted to go through the calculus to find the exact (well kinda) answer. In the end, probably due to some of the assumptions I made, I was off by a factor of 2.
In this video, I attempt to find the top speed of the Saturn V rocket if it were already free from earth's gravity and fully fueled.
Second we create an interesting shape by cutting into the side of a cylinder. Almost 10 years ago when I was still a university student on an internship, one of the drafters asked me what the volume of a circular cut out of the side of a cylinder would be. He already knew the answer, but I guess he wanted confirmation. I wasn't so great at calculus at the time and I didn't have any 3D software to help me visualize the shape so it took me three days to arrive at the answer. In the end I was surprised at how easy it was when you looked at the shape the right way.