The Fundamental Theorem of Calculus
Using the FTC to Evaluate Integrals
If f is the derivative of F, then we call F an antiderivative of f.
We already know how to find antiderivatives–we just didn't tell you that's what they're called. It's like when you realize what all of the subtle signs in the M. Night Shyamalan movie mean. Seriously, like whoa. Whenever we are given a derivative and we "think backwards" to find a possible original function, we're finding an antiderivative.
Let f(x) = 3x2. Find an antiderivative of f.
We think backwards: what could we take the derivative of to get 3x2? This derivative looks like it came from the power rule, so the original function must involve x3. Since the derivative of x3 is 3x2, the function
F(x) = x3
is an antiderivative of f(x) = 3x2.
Any other antiderivative of 3x2 will have the form x3 + C where C is a constant. We generally take C = 0. For the FTC it won't matter which antiderivative we use, so we might as well use the simplest one.
These exercises should be mostly review, and help you remember how thinking backwards works. You might want to review the rules for taking derivatives first.
To check an answer for this sort of problem, take the derivative of your answer. If you take the derivative of your answer F and get the f given in the problem, then F is an antiderivative of f and you did the problem correctly. Gold stars all around.
Now that we know what antiderivatives are, we can use them along with the FTC to evaluate some integrals we didn't know how to evaluate before. The FTC says that if f is continuous on [a,b] and is the derivative of F, then
This means if we want to know , we
1) find an antiderivative F of f,
2) evaluate F at the limits of integration, and
3) subtract to find F(b) – F(a).
When evaluating definite integrals for practice, you can use your calculator to check the answers. If you don't know how to use your calculator to find integrals you can look in the manual, look online, ask a friend, or ask your teacher. But practice doing integrals by hand until they're so easy you don't even mind anymore.
Here are some reasons to practice doing integrals by hand.
1) At some point you'll probably need to pass a test involving integration, without being allowed to have a calculator. Midterm, anyone?
2) Even when you are allowed a calculator, your teacher will probably want to see the steps you took to get your answer.
3) If you're asked to integrate something that uses letters instead of numbers, the calculator won't help much (some of the fancier calculators will, but see the first two points).
4) Later in Calculus you'll start running into problems that expect you to find an integral first and then do other things with it. It will sometimes be easier to find the integral by hand than it will be to distract yourself by putting the integral into your calculator.
Before asking you to find too many definite integrals, we should share a nice notational shortcut. To abbreviate
F(b) – F(a),
This expression is read "F of x evaluated from a to b" (have button - click to hear pronounced).
Using this shortcut, our work to find would look like this:
This is a nice shortcut because it saves you from having to mess with letters like f and F. You just find the antiderivative
put it in brackets
draw a vertical line on the right-hand side
and write the limits of integration
Then you expand the shortcut
[(2)3] – [(0)3]
and simplify to get the answer.
Remember that when you expand the shortcut, you use the upper limit of integration first: