Integrals As Signed Areas Under a Curve
Graphically speaking, an integral is the signed area under a curve. So, from the interval A to B on the graph above, the highlighted region is the integral. This graph is the same as the one up there, except that the axes are now labeled.
To integrate on the interval A to B, we must find the area of the red triangle. If we do that, we'll end up with an equation similar to the one in the green box. Don't mind the "1/2". What's important to note is that if we solve this equation, the s's will cancel, and we'll be left with the unit m, as seen in the purple box. As we know, "metres" is a unit that indicates "position", d. The parent graph of a velocity-time graph is a position-time graph. YAY! This shows what we learned about integrals: that they are signed areas under a curve, and that they are the total change on a parent function.
I'll explain the last concept we learned as quickly as possible. The graph above is, again, the same as the ones before, except that the integral from A to B is now divided into a blue region and green region. We learned that if we find the integral from A to Z and add that to the integral from Z to B, we'll end up with the same number as if we just took the integral from A to B.
That was the class!
Hey guys, hope you had a great winter break. Back to school now! No one skip the first day of classes! Can't wait to see all of you =)
I guess we're starting the new cycle, so the next scribe is... Crystal!