121616949-math.163

121616949-math.163 - namely Z 2 1 x 2 dx = 2 3 3-1 3 3 = 7...

This preview shows page 1. Sign up to view the full content.

7.2 The Fundamental Theorem of Calculus 149 hard to prove. Once the last reference to interpretation has been removed from the proofs of these facts, we will have a real proof of the Fundamental Theorem. Now we know that to solve certain kinds of problems, those that lead to a sum of a certain form, we “merely” find an antiderivative and substitute two values and subtract. Unfortunately, finding antiderivatives can be quite difficult. While there are a small number of rules that allow us to compute the derivative of any common function, there are no such rules for antiderivatives. There are some techniques that frequently prove useful, but we will never be able to reduce the problem to a completely mechanical process. Because of the close relationship between an integral and an antiderivative, the integral sign is also used to mean “antiderivative”. You can tell which is intended by whether the limits of integration are included: Z 2 1 x 2 dx is an ordinary integral, also called a definite integral , because it has a definite value,
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: namely Z 2 1 x 2 dx = 2 3 3-1 3 3 = 7 3 . We use Z x 2 dx to denote the antiderivative of x 2 , also called an indeﬁnite integral . So this is evaluated as Z x 2 dx = x 3 3 + C. It is customary to include the constant C to indicate that there are really an inﬁnite number of antiderivatives. We do not need this C to compute deﬁnite integrals, but in other circumstances we will need to remember that the C is there, so it is best to get into the habit of writing the C . When we compute a deﬁnite integral, we ﬁrst ﬁnd an antiderivative and then substitute. It is convenient to ﬁrst display the antiderivative and then do the substitution; we need a notation indicating that the substitution is yet to be done. A typical solution would look like this: Z 2 1 x 2 dx = x 3 3 ﬂ ﬂ ﬂ ﬂ 2 1 = 2 3 3-1 3 3 = 7 3 . The vertical line with subscript and superscript is used to indicate the operation “substitute and subtract” that is needed to ﬁnish the evaluation....
View Full Document

{[ snackBarMessage ]}

What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern