How to solve integrals, well...kinda...and what they can mean. Last year when we learned it at the end of the year, I almost failed every single quiz and test on it. Now, I actually get it. ^.^ I'll post some of what we did in class... It explains a lot.
Words that also mean "integral" or are associated with integrals: Area [under the curve], antiderivative, accumulation (and "Awesome", according to my calculus teacher
Example AP Calculus AB Problem:
A test plane flies in a straight line with positive velocity.
Use a midpoint Riemann Sum with 4 subintervals of equal length to approximate
∫ v(t)dx from a=0 to b=40
Then we solved it...
First we found the Length of the Subintervals by using the equation: (b - a)/n, with n representing the number of subintervals. The length of each subinterval turns out to be 10.
The subintervals, based on the above chart, are as following: from 7.0 to 9.5, from 9.5 to 4.5, from 4.5 to 2.4, and from 2.4 to 7.3.
The "midpoint Riemann Sum" method simply means you take the the middle value of those subintervals (provided on the chart) and use them when you solve for the area of each rectangle you draw in the sketched graph (sketching the graph is optional, but we did it anyway).
We drew a sketch of the graph using the midpoint Riemann Sum method too... This is what it looked like:
Then, we solved for the area of each of those rectangles and added them together in order to find the approximate value of the area.
(10)(9.2) + (10)(7.0) + (10)(2.4) + (10)(4.3)
= 92 + 70 + 24 +43
But since we're solving for an integral
and not just the value of the area, we need correct units and to write what this value we found means.
So, since we're multiplying the time by the velocity we have:
And we have "miles" as our units.
This means that wha we solved for is as follows:
The plane flew 229 miles from t = 0 to t = 40 (or, the plane flew 229 miles in 40 minutes).
And that is what I learned today that I did not know yesterday
We also learned some other rules for integrals, but those are rather hard to type up here (yes, harder than what I just posted), and so I shall refrain from doing so. Besides, you're all probably sick of calculus now 'cause of this.