A few intro calculus ideas to help explain why we study basic properties of sums

My older son is doing a some review this summer in the Integrated CME Mathematics III book this summer. The topic in the 2nd chapter of the book is sequences and series. I thought it would be fun to show him where (at least some of) this math leads. So tonight we talked about some basic ideas in calculus.

First I introduced the topic and reviewed some of the basic ideas of sequences and series:

Now we used the ideas from the first part to find the area under the curve y = x by approximating with rectangles:

To wrap up we extended the idea to find the are under the curve y = x^2 from x = 0 to x = 2. It was fun to see that the basic ideas seemed to makes sense to him.

I was really happy with how this project went. Putting these ideas together to calculate the area under a curve – even a simple curve – is a big step. It might be fun to try a few more examples like these before moving on to the next chapter.