Calculus: The Key Ideas (9/3/08)

What are the key ideas of calculus? Let’s review and discuss.

Today we review the concepts of definite integral, antiderivative, and the Fundamental Theorem of Calculus (which tells how the first two concepts are related).

The Definite Integral

What does it mean to “integrate a function” over some part of its domain?

Answer: To add up its values on that part of the domain.

How can you do this?? Partial Answer: If it has only finitely many

values, just add them up, giving each value appropriate “weight.” (Such a function is called a “step-function”.)

The Integral Continued

What if (as is usually true) our function has infinitely many values on the part of the domain of interest?

First Answer: We can estimate its integral by using a finite number of values and giving each an appropriate “weight”.

We can get better estimates by using more and more values over shorter and shorter intervals.

The Integral in Symbols

If we seek the integral of f on the interval [a,b], we can estimate it by taking n equally spaced points along the interval: a = x0 < x1 < …< xn-1 < xn = b. The little intervals between these are of length x = (b-a)/n.

Our estimate could then be:f(x0) x + f(x1) x +…+ f(xn-1) x (“left-hand sum”)or

f(x1) x + f(x2) x +…+ f(xn) x (“right-hand sum”).

The average of these two is called the Trapezoid Rule.

Getting it exactly right

How can we turn this estimate into the exact integral?

Answer: Take the limit as n !! Another (very cool) answer: Suppose we can

view our function f as the rate of change of another function F. Then the integral of f over [a,b] will simply be the total change in F over [a,b], i.e., F(b) – F(a). This is called the Fundamental Theorem of Calculus.

The Fundamental Theorem (Part I)

We are given a function f(x) on an interval [a,b] If F(x) is any antiderivative of f(x), then

That is, to “add up” f ’s values from a to b, it suffices to find an antiderivative of f (not necessarily an easy thing to do!!), evaluate it at the endpoints, and subtract.

b

a

aFbFdxxf )()()(

Recall Some Antiderivative Facts

x r dx = (1/(r +1)) x r +1 + C unless r = -1(The “Reverse Power Rule” – “push up, divide”)

1/x dx = ln(x) + C a x dx = (1/ln(a)) a x + C loga(x) dx = ?????? sin(t) dt = - cos(t) + C cos(t) dt = sin(t) + C tan(t) dt = ??????

Assignment for Friday

Be sure to have the text and always bring it to class (except labs).

Review Sections 4.9 (Antiderivatives) and Section 5.1-5.3 (Integrals and the Fundamental Theorem) as needed to be sure you’re up the speed.