Definite Integrals

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What is the “x_i*” in the formula for a definite integral?

It’s a “typical” or “representative” x value in the i^th interval in the Riemann sum. Each interval corresponds to a rectangle whose area is being used to approximate the area under a small portion of a function’s graph. The height of that rectangle is the function’s value at a representative x value in the interval; that value is x_i*. You can pick any representative value you like in the interval; the left side or right side are common choices.

Misc

SI

The next session is tonight from 6:30 - 8:00. It will review today’s class and then have time to talk about problem set questions.

Remote Learning Plans

After Thanksgiving, this course will still revolve around readings, discussions, problem sets, and individual meetings, with periodic Zoom “classes” to collect answers to discussion questions. The Zoom “classes” shouldn’t introduce new material, but should rather serve to summarize questions and ideas from discussions and readings into coherent documents that will be posted in Canvas.

SI sessions will also continue after Thanksgiving, although on a different schedule. SI sessions continue to be reviews of what you learn through readings, discussion, and Zoom “classes,” not replacements for those things.

There is class next Monday (November 23), sticking to the regular cohort schedule (it will be a Cohort C day) even though Monday is following a Wednesday class schedule.

Colloquium

“Making Pooling Designs for COVID-19 Surveillance on a Mass Scale Deployable and Decodable in the Field”

Prof. Anthony Macula, SUNY Geneseo.

Thursday, November 19, 4:00 - 5:00 PM.

Discusses some of the math behind fast-turnaround, efficient, COVID testing, such as the pooled tests many of you presumably have done or are doing.

Definite Integrals

From “Definition and Notation,” “Evaluating Definite Integrals,” “Area and the Definite Integral,” and “Properties of the Definite Integral” in section 5.2 of the textbook, and this discussion of definite integrals.

Properties of Definite Integrals

Combine knowledge of simple integrals to evaluate more complicated ones.

In particular, here are some facts about some integrals:

Given these facts, what is the integral from 1 to 2 of x²?

Use the property that you can break integrals over one interval into sums of integrals over sub-intervals (this property basically comes from the fact that integrals are sums and sums are associative, i.e., you can group the numbers being summed together any way you want):

What about the integral from 0 to 2 of 3x²?

Use the constant multiple rule for integrals to factor the 3 out of the integral (this rule comes from the fact that you can factor a common coefficient out of a sum):

Finally, how about the integral from 0 to 2 of x² + x?

This time, there’s a sum rule for integrals you can use (this comes from associativity of addition again):

Integrals and Area

What do you think the integral from 0 to 2π of sin x is?

Because integrals are signed areas (i.e., positive area under a graph that’s above the x axis, but negative area above a graph that’s below the axis), and sine is symmetric, you would expect the integral to be 0. As it is (we won’t prove that yet).

Do Riemann sums “get it right” when there’s area below the x axis as well as above? For example, what should the integral from 0 to 3 of x - 1 be, and does a Riemann sum get the same answer?

Geometrically, the area under the graph and above the x axis, minus the area below the axis, works out to 1 1/2 square units.

Start by writing out the integral as a Riemann sum.

This isn’t all that informative until you fill in expressions for Δx and f(x_i*). Since the region to integrate over is 3 units long, and we’re dividing it into n intervals, each interval is Δx = 3/n units long. And if we use the right edge of each interval as its x_i*, then x_i* is just i Δx = 3i/n:

Plugging these values into the Riemann sum formula, we get...

Now we can use algebraic rules for sums, and closed form formulas, to rewrite the sum. Notice in particular here that relative to “i”, the thing that varies from term to term in summation notation, “n” is a constant that can be factored out of sums. Also notice that the sum from 1 to n of a constant (e.g., 3/n) is that constant times n:

Finally, simplify the expression inside the limit and use laws about limits at infinity to get a number:

Next

We need a better way to evaluate integrals than Riemann sums. Fortunately, something called “the Fundamental Theorem of Calculus” provides it.

So to get started on the Fundamental Theorem, please read “Average Value of a Function” in section 5.2 of the textbook, and the introduction, “The Mean Value Theorem for Integrals,” and “Fundamental Theorem of Calculus, Part 1: Integrals and Antiderivatives” in section 5.3 by class time Thursday.

Please also participate in this discussion of part 1 of the Fundamental Theorem by class time Thursday.

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