![]() Here are some reasons to practice doing integrals by hand.ġ) At some point you'll probably need to pass a test involving integration, without being allowed to have a calculator. But practice doing integrals by hand until they're so easy you don't even mind anymore. 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. When evaluating definite integrals for practice, you can use your calculator to check the answers. ![]() The FTC says that if f is continuous on and is the derivative of F, thenĢ) evaluate F at the limits of integration, and 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. 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. To check an answer for this sort of problem, take the derivative of your answer. You might want to review the rules for taking derivatives first. These exercises should be mostly review, and help you remember how thinking backwards works. For the FTC it won't matter which antiderivative we use, so we might as well use the simplest one. Since the derivative of x 3 is 3 x 2, the functionĪny other antiderivative of 3 x 2 will have the form x 3 + C where C is a constant. ![]() We think backwards: what could we take the derivative of to get 3 x 2? This derivative looks like it came from the power rule, so the original function must involve x 3. Whenever we're given a derivative and we "think backwards" to find a possible original function, we're finding an antiderivative. It's like when you realize what all of the subtle signs in the M. We already know how to find antiderivatives–we just didn't tell you that's what they're called. Just as we integrate velocity to find the change in position, we integrate the rate of change of f to find the change in f.If f is the derivative of F, then we call F an antiderivative of f. If you're getting mixed up with all the f's and F's, you can think of the FTC as Or, since the important part of this equation is that the function being integrated is the rate of change of the function outside the integral, we could also write Since we can change the variable of integration, we could write Remember thatģ) on the right-hand side, we use the upper limit of integration first: Being able to write downįrom memory is a good start. You should memorize the FTC, but you should also understand what it means. The Fundamental Theorem of Calculus (FTC): If f is continuous on the interval and is the derivative of F on, then We can generalize from the special case with position and velocity to get the following. Is the rate of change, or derivative, of the function outside the integral As far as the mathematics is concerned, the important thing is that the function inside the integral There's no particular reason that s( t) needs to describe position and v( t) velocity. ![]() Either of these quantities gets us the net change in s from a to b. Is the weighted distance travelled, taking into account that distance travelled in opposite directions cancels out. Is the difference between the starting position at t = a and the ending position at t = b, while If s( t) is a position function with rate of change v( t) = s'( t), thenīecause both of these quantities describe the same thing:
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