SUNY Geneseo Department of Mathematics

The Fundamental Theorem of Calculus, Part 1

Thursday, November 19

Math 221 02
Fall 2020
Prof. Doug Baldwin

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Anything You Want to Talk About?

Riemann sums?

Basically, a Riemann sum is a formal mathematical description of adding the areas of rectangles to approximate the area under a graph. The idea starts by dividing the part of the x axis over which you want the area into intervals, each of width Δx. This is the width of the rectangle in each interval. The height of the rectangle is the function’s value at some “representative” or “typical” place in the interval, xi* (read this notation as “x” because the variable represents an x value, “i” because it’s the x value associated with the ith interval, and “*” to indicate that it’s the representative value for that interval). Now the area of a rectangle is its width times its height, thus the product in the Riemann sum, and you add up the areas of all the rectangles to get the total (estimated) area under the curve, thus the sum:

Area under graph divided into rectangles of width Delta x and height f of x sub i star

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.

Flow diagrams: 'reading/discussion' leads to 'Zoom' and back, 'problem set' to 'meeting' and back, 'SI meetings' loop

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.

Is Zoom the best technology for you? Let me know, especially if not.

The Fundamental Theorem, Part 1

Based on “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, plus this discussion of part 1 of the Fundamental Theorem.

The Functions

For example, define F(x) to be the integral from 0 to x of t:

F of x equals integral from 0 to x of t

How do you evaluate F? Is there an equivalent “closed form” (i.e., definition without the integral)? What is it? Is its derivative what the Fundamental Theorem says it should be?

Geometrically, F(x) is the area of a triangle whose base extends x units along the x axis:

Straight-line graph with triangle with base x below it

Since the height of the triangle is also x (since the top edge is the line y = t), the area formula for triangles gives the area, i.e., the value of F(x) for any x. Use this to calculate some sample values:

Use area formula base times height over 2 to calculate F of x

These examples generalize to a closed form for F(x):

F of x equals x squared over 2

Differentiating the closed form via the power rule, and the integral form via the Fundamental Theorem, do indeed produce the same result:

Derivative of integral to x of t equals derivative of t squared over 2

Applications

Use the Fundamental Theorem to find some derivatives...

These two examples demonstrate the simplest, most direct, uses of the Fundamental Theorem, simply plugging a new variable into the integrand. Notice that it doesn’t matter what the lower bound for the integral is.

Derivatives found by replacing variable in integrands

If the variable is in the lower bound rather than the upper bound, use the fact that switching the bounds on an integral just switches its sign to put the variable in the upper bound and then use the Fundamental Theorem:

Negate integral and switch bounds to use Fundamental Theorem

If the upper bound on the integral is a function, then use the chain rule to find the derivative:

Derivative of integral to sine y of t squared over log t is sine squared of log of sine, times cosine

Significance

Part 1 of the Fundamental Theorem ties antiderivatives and definite integrals together by showing that any antiderivative is also a definite integral. It unifies antiderivatives (and thus derivatives) and integrals into “calculus” instead of separate subjects.

Next

Another relationship between integrals and antiderivatives: the Fundamental Theorem Part 2.

Please read “Fundamental Theorem of Calculus, Part 2: The Evaluation Theorem” in section 5.3 of the textbook by class time Friday.

Please also participate in this discussion of the Fundamental Theorem part 2 by class time Friday.

I’ll also try to fit in how to find integrals and antiderivatives with Mathematica.

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