r/calculus

Can someone please explain how to do these questions in details? Really appreciate it!

Did this one with no hints so it seems like studying the gamma/zeta functions is paying off

I am primarily a visual learner and sycamore.to has been great to help me understand complex math concepts visually.

For preface, I am a high school Calculus student. We just finished our first unit on integrals. During my study period today, I started thinking about what happens when you take the area between two curves and rotate it about both axis. I attempted to find a method to do this intuitively.

My hypothesis is, I can find the volume through the following equation:
V = Vx + Vy - delta V

I was able to prove the volume of the shapes created by just rotating on the X axis and just rotating on the Y axis. But I can’t confirm that my method of calculating delta V actually represents the geometric intersection of both solids.

I also noticed that delta V is greater than Vy, which doesn’t feel right if it truly represents the overlap.

Am I close? Are there any known methods for something like this? Is there any feedback you could give on whether the delta V step holds up geometrically and what it could represent if it isn’t the overlap?

EDIT:
I uploaded a second image however it seems to have not loaded. Essentially in it I calculate delta V by subtracting Vy from Vx and then apply my hypothesized formula. I got a result of 9 pi units cubed.

So, basically, this hall building is gonna be demolished this summer. So, I grab plenty of books from there including these two books. The book on the right was grabbed yesterday. The book on the left was grabbed 7 days ago. Old textbooks are so cool as references.

I was teaching the very beginnings of the idea of how to find the slope of a tangent line at (1,1) for the function f(x) = x^2 last Thursday.

We did it by 'creeping' up to x = 1 from both ends, like.. Idk it's hard to describe without my mouth or the ability to write on a whiteboard.

"We could hold the point (1,1) fixed and 'drag' the point (3, 9) to the left towards (1,1) until we got really close, like (1.01, 1.0201), and find the slope between those two points. That would be a really good approximation."

*makes table with x = 1.2, 1.1, 1.01*

"We could do the same from the under-side as well. Hold (1,1) fixed and slide (0,0) up and right really close to (1, 1), like (0.99, 0.9801) and find that slope."

Q: What does it appear the slope should be?

Class A: "It looks like the slope should be 2!

But then a student of mine asked a fun question: What if we took a point on either slide and slid both of those towards (1,1)? Would that work?

I had to pause for a bit and think about it. My answer was that having two moving parts might be a bit more cumbersome than needed, but yes it should work.

A few days later (today), the thought popped back into my head, so I did it. It works and it's simple and it's cool.

Recently got my script-to-manim animation platform (tensorframes.co) to handle long form videos. Feedback welcomed.

Check out more animations @tensor_frames

im doing some last minute review for mcv4u (Ontario calc and vectors), and I came across this question and im not sure what to do, could someone help?

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