DOC PREVIEW
MIT 18 01 - PROPERTIES OF INTEGRALS

This preview shows page 1 out of 3 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 3 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 3 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable CalculusFall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.PI. PROPERTIES OF INTEGRALS For ease in using the definite integral, it is important to know its properties. Your book lists the following' (on the right, we give a name to the property): (1) f () dx = -abf(f(s)da integrating backwards (2) f(T)dz =0 (3) f (x)dz = f(z)dx + jf(z)dz interval addition (4) +g)d = f()d + 9g(x)d linearity l f(z)dz = c f ()dx linearity f(z)d < jg(z)dls if f() g() on [a, b estimation Property (5)is useful in estimating definite integrals that cannot be calculated exactly. Example 1. Show that r-dz < 1.3. Solution. We estimate the integrand, and then use (6). We have z3 < X on [0, 1]; e VI w = (1+ )o = (2%/i-1) m1.22 < 1.3 . We add two more properties to the above list. (Q) if(x)dxl :5 f(x)j d . absolute value property Property (6) is used to estimate the size of an integral whose integrand is both positive and negative (which often makes the direct use of (5) awkward). The idea behind (6) is that on the left side, the intervals on which f(x) is negative give a negative value to the integral, and these "negative" areas lower the overall value of the integral; on the right the integrand has been changed so that it is always positive, which makes the integral larger. Example 2. Estimatethesize of e-sin dz. 1m Sinmoms pp. 214-215 . ..r· ·. ~ · PI. PROPEFTIES OF INTEGRALS Solution. A crude estimate would be e-'sinz dxj :_] e-x'sinzl•Io 100 < e-"dz, by (5), since Isinsxj 1;]00 =--e-0 -e 10 +1 < 1. A final property tells one how to change the variable in a definite integral. The.formula is the most important reason for including dz in the notation for the definite integral, that is, writing f(z) dz for the integral, rather than simply f (z),as some authors do. Id * , = u( d), (7) f(u)du = f(u()) dx, c = U(a), change of variables formula t = u(b). In words, we can change the variable from u to z, provided we (i) express du in terms of dx; (ii) change the limits of integration.2 There are various possible hypotheses on u(x); the simplest is that it should be differen-tiable, and either increasing or decreasing on the x-interval [a, b]. Example 3.Evaluate (1+ue) /2 by substituting u= tanx. Solution. For the limits, we have u = 0,1 corresponding to z = 0, /4; tan x is. increasing. I dur l sec2 z +(1+ 2)s/2 j= sec a w/4 1/4 -cos dxs = sin] = -N. Proof of (7). We use the First Fundamental Theorem3 and the chain rule. Let F(u) be an antiderivative: (8) F(u) = f(u) du; d dF du du F(U()) = -= f (uu) , by the chain rule. So (9) )) = f ( )) d. Therefore f(u) du = F(d) -F(c), by the First hdamental Theorem and (8); = F(u(b)) -F(u(a)) = f(u(x)) ds, ' by the First •undamental Theorem and (9). D asee Simmons p.839 for a discussion and an example 3aee the naxt


View Full Document

MIT 18 01 - PROPERTIES OF INTEGRALS

Documents in this Course
Graphing

Graphing

46 pages

Exam 2

Exam 2

3 pages

Load more
Download PROPERTIES OF INTEGRALS
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view PROPERTIES OF INTEGRALS and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view PROPERTIES OF INTEGRALS 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?