Introduction to Calculus/Integration

We all know that you can take a derivative to find the slope of a function, but what if you wanted to use the derivative to find the original function? That's where integration comes in!

Okay, so number 1. Let's talk about Riemann Sums.

Essentially, a Riemann Sum is an approximation of area under some curve. Let's say you had the function y=2x. What if you wanted to find the area between the curve and the X-axis? I mean, when I learned this, I was just like "lol why would we need to know this" BUT I then learned that it is SUPER useful when determining areas of objects, and if you're continuing in math, it'll be really cool to get this down. Okay so back to the curve y=2x. If you made 4 rectangles under the curve to find the area between x=0 and x=4, you might be able to use the left points, or L 4, to approximate a lower amount then the curve's actual area (assuming that the graph is concave up, maybe somebody will write about that later). That is called an underestimate.

This is what that would look like: L 4 = ᐃx*(f(0)+(f(1)+f(2)=f(3)) = 1*(0+1+4+9) = 14

Another fun thing you can do is get an overestimate by having those 4 rectangles maintain a Delta X of 1, given this interval, but instead, you use the right side of each value. If we had 4 rectangles to estimate it, then our delta x would be equal to (domain of x)/(number of values), so in this case, 1/1 = 1. That Riemann Sum would look like this:

R 4 = ᐃx*(f(1)+(f(2)+f(3)+f(4))

Oh I have to go help my mom bring in groceries but anyways thank you all for reading this! Sorry if the writing was really bad and also that this page was unfinished. I'm 15 and I just felt bad that there was nothing on this chapter yet. Also I took calculus two years ago so I might have been a little rusty. Bye!