Essential Knowledge 3.2A2
Essential Knowledge 3.2A2 Students will know that the definite integral of a continuous function f, over the interval [a, b], denoted by , is the limit of Riemann sums as the widths of the subintervals approach 0. That is,
where xi* is a value in the ith subinterval, and Δxi is the width of the ith subinterval, n is the number of subintervals, and max Δxi is the width of the largest subinterval. Another form of the definition is
, where and xi* is a value in the ith subinterval.
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Time to take the final step. So far we've gotten to the point where this is our formula:
Since you know our end goal is to create infinitely many rectangles, you might think we'd do this
and you'd be close. The problem is
would become zero. We want our subintervals to become extremely small and thus approach zero, but not actually be zero. So of course, when you hear things like "approach zero", who do we know that approaches things? Especially zero and infinity? Our old friend limits!
So instead of setting n to infinity, we'll let it approach infinity, like so
For me this is good enough, but there are a few things College Board apparently wants you to know for this test.
They like 2 formulas. Here's one
Same thing as the one we made, except xi has a little * on it now. This star means that xi doesn't have to be on the left endpoint of the subinterval. It can be on the left, right, or anywhere in between. This makes sense to me, because the subintervals are getting infinitely small, which means the left and right endpoints are getting infinitely close to each other, and thus they'll essentially be the same point.
The other one is this
In this one, instead of deciding on the number of rectangles, then using that to calculate the length of each interval, we decide on the length of the interval, and use that to calculate the the number of rectangles. Whatever!
And now that we have our equation, it's time to turn it into an integral. Here's the statement from the Essential Knowledge, because it says it well enough:
, where and xi* is a value in the ith subinterval.
The only thing I haven't shown you get is how to use the a and b. Integrals that have an a and b are called a definite integral. Integrals that don't have it and therefore whose answers have a + C are called indefinite integrals.
But we can worry about that later. For now, what's important is that you have seen how we have taken the process of counting rectangles and turned it into the integral notation.