Chapter 18: The Fundamental Theorem

where u is the variable and x is only a " dummy variable " ( variable of integration ). This F(u) is simply the " area function " we defined before (except that we called it A(u) rather than F(u) in the last section). In the last section we showed that the derivative of this area function is f(u). Thus, taking the derivative of F returns us to the original function f that we integrated to get F. In words, the fundamental theorem says that if a function is defined by taking the definite integral from a fixed lower limit of integration up to a variable upper limit of integration, then its derivative is obtained by simply dropping the integral sign and taking the integrand with the dummy variable replaced by the variable in the upper limit of integration ). In symbols,

F(u) = = f(u).

Example 1: Find of the function defined as

a. ;
b. ;
c.

In parts (a)--(b), check your answer by finding a formula for F(u) and computing the derivative of this formula.
(a) By the fundamental theorem, d/du Int(x^2,x=0..u) = u^2. Let us check this by finding = = . Since ( ) = , the answer checks.
(b) By the fundamental theorem, d/du , the same answer as in part (a). But our function F(u) is not quite the same. Namely,

F(u) = = = - .

Of course, this F(u) still has derivative . This example illustrates the fact that the derivative of a function of the form does not depend on what constant value we choose for a , the lower limit of integration.

(c) By the fundamental theorem, d/du = . Here we cannot find a formula for F(u), since we do not know how to find the anti-derivative of a complicated function like . But thanks to the fundamental theorem, if all we want is the derivative of F(u) (and not F(u) itself), we never have to worry about anti-derivatives. The derivative of F(u) can be determined right away without any work.

The Fundamental Theorem together with the Chain Rule

Suppose that u is a function of some other variable, like time, and we want to find

= .

This can be done using the chain rule:

= .

Example 2: Suppose that in Example 1(c) we have , and we want to find the time derivative of the integral. Then

= = = .

If the function is defined by an integral in which occurs in the lower limit of integration , then simply reverse the upper and lower limits, and put in a minus sign, i.e., (see property (3) of definite integrals).

What if a function appears in both limits of integration, i.e., what if you want to take of , where u and v are each functions of t? Then choose a constant c and use properties (2) and (3) of definite integrals with a replaced by u and b replaced by v:

+ -

We already know how to take the derivative of each of the two terms on the right. Thus,

= -

For example =

Example 3: Suppose that the curve forms the top of a shape that is bounded below by the x axis and on the left and right by vertical lines whose position changes with time. Suppose that the vertical line on the left is moving steadily to the right at speed 0.1, starting at -1 at time t = 0. Meanwhile, the vertical line on the right is oscillating between 1 and 3 according to the formula . Find a formula in terms of t for the rate of change of the area of the shape between the two vertical lines.

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