The Fundamental Theorem of Calculus - The Second Fundamental Theorem of Calculus

The Second Fundamental Theorem of Calculus

As if one Fundamental Theorem of Calculus wasn't enough, there's a second one. The first FTC says how to evaluate the definite integral

if you know an antiderivative of f. The second FTC says how to build an antiderivative of f if you don't know one already.

Given a continuous function f and a constant a, we can define a new function F(x) by integrating f from a to x:

We can think of this new function F as an accumulation function—it accumulates the weighted area under f from a to x.

Given a continuous function f, we can build a new function F(x) by picking a lower limit of integration a and integrating f from a to x.

The Second Fundamental Theorem of Calculus says that when we build a function this way, we get an antiderivative of f.

Second Fundamental Theorem of Calculus:

Assume f(x) is a continuous function on the interval I and a is a constant in I. Define a new function F(x) by

Then F(x) is an antiderivative of f(x)—that is, F '(x) = f(x) for all x in I.

That business about the interval I is to make sure we only get limits of integration that are are reasonable for your function. Some things that wouldn't be reasonable:

• Choosing a = -1 for the function
  . Then
  
  doesn't make sense because if x ≠ -1 we're trying to integrate a function over an interval where the function isn't even entirely defined.
    
• Choosing a = 1 for the function  but then trying to evaluate
  
  Again, we're trying to integrate a function over an interval where the function isn't entirely defined.
    
• Choosing a = 0 for the function . How would you evaluate
  
  The problem is that a = 0 isn't in an interval on which f is continuous. Actually, we can do stuff with this integral—it just requires some extra work, and we haven't gotten there yet.
    
• Choosing a = -1 for the function  and trying to evaluate
  
  If I is any interval that contains both -1 and 1, the function  isn't continuous on that interval. Again, we can do stuff with this integral, but we haven't learned how yet.

We can condense this discussion into a short warning.

Be Careful: Whenever you go to evaluate an integral, make sure the limits of integration are reasonable for the function you're integrating.

If a function is discontinuous at or between the limits of integration we may still be able to evaluate the integral, but it requires a little extra sneakiness that we haven't gotten to yet.

So far, we've used the second FTC to build antiderivatives for things we already have derivatives for. That's not very exciting. Why go to all the trouble of writing

to get an antiderivative for sin x, when we already know that

-cos x

is an antiderivative of sin x?

It turns out there are many functions whose antiderivatives we don't know how to find. Some we'll learn how to find using various techniques of integration. However, there are many functions whose antiderivatives can't be written down with a nice formula. We can't write down a nice formula for the antiderivative of

e^ex

or

,

for example - not just because we don't know how, but because it can't be done.

The second FTC lets us build an antiderivative of f(x) even in the cases where it's impossible to write a nice formula for an antiderivative.

• Letters

  The second FTC can be confusing with all those letters x and t floating around. Chillax. We're going to straighten them out here.

  A function and its derivative use the same independent variable. If we have a function f(x) and we take its derivative, we get a function f '(x). We don't get f '(t).

  Working backwards, a function and its antiderivative also use the same independent variable. If we're trying to build an antiderivative for f(x), the antiderivative must also be a function of x. So we need to end up with an antiderivative F(x), not F(t). This is why we use x as a limit of integration: when you change the upper limit of integration, you change the value of the definite integral.

  When x is a limit of integration, we don't want to use it as the variable of integration too. Writing

  

  is horribly confusing. Other than that, it doesn't matter what letter we use for the variable of integration. We could write

  

  or

  

  or

  

  When we change the letter, all we're doing is relabeling the horizontal axis. Regardless of the labeling, we're finding the weighted area between f and the horizontal axis from a to x.

• Antiderivatives

  The FTC tells us how to build antiderivatives for any continuous function we like. No user manuel required. There are some things we need to know about these antiderivatives.

  If we define an antiderivative of f(x) by

  

  then

  

  because we're integrating over an interval of length 0.

  1) If we define two antiderivatives of f(x) by

  

  

  where a < b, then the difference between F_b(x) and F_a(x) is

  

  which is a constant.

  2) Define an antiderivative of f(x) by

  

  Since the derivative of a constant is 0, we can add any constant c that we like to F(x) and get a new antiderivative of f(x). That is, the function

  

  has G '(x) = f(x).

  Using the information above, we can build antiderivatives with various properties.

• Finding Derivatives

  The second FTC states that a function of the form

  

  is an antiderivative of f(x). This means F '(x) = f(x). Using Leibniz notation,

  

  Sample Problem

  Find
  

  Answer.

  The second FTC says that the integral expression

   is an antiderivative of e^ex. This means the derivative of  is e^ex. In Leibniz notation,

  

  In the previous example, all we had to do was write down the integrand, replacing t with x. This works so long as the lower limit of integration is a constant and the upper limit of integration is x all by itself.

  We could also change the letters around. If we're using t as the upper limit of integration, then we have an antiderivative F(t) and its derivative will also be a function of t.

  Sample Problem

  

  Answer.

  This question is asking for the derivative of

  

  Since F is a function of t, its derivative is also a function of t. All we have to do is write down the integrand, replacing x with t so that we end up with a function of t.

  

  Using properties of integrals and the chain rule, we can find the derivatives of more complicated functions.