Week 13 Calculus I Outline of Notes

by S. Gramlich

(updated 12/11/06)

I) 5.4 An indefinate integral has NO limits of integration, whereas a definate

integral has limits of integration.

An indefinate integral is a family of functions so remember to add a constant after you evaluate; whereas a definate integral is a single # after evaluation.

p. 406, Table of Indefinate Integrals

5.4, p. 411, #8

5.4 Applications: Net Change Thereom

_a∫^b F'(x) dx = F(b) - F(a) is net change from a to b

Read pp. 408-409

ie. _t1∫^t2 v(t) dt = _t1∫^t2 s'(t) dt = s(t₂) - s(t₁) is net change of position (displacement) of a particle during time period from t₁ to t₂.

Likewise, _t1∫^t2 a(t) dt = _t1∫^t2 v'(t) dt = v(t₂) - v(t₁) is net change of velocity of a particle during time period from t₁ to t₂.

II) 5.5 Substitution Rule = like converse of chain rule

When can't use antidifferentiation formula, make a "u" subsitution

1) Let u = some part of the integrand

2) differentiate du/dx (u wrt x)

3) manipulate step 2 literally in terms of dx

4) Substitute u & step 3 result

5) integrate using an antidifferentiation formula

6) Resubstitute back in for u

Idea is to replace a relatively complicated integral by a simpler integral.

∫' f(g(x))g'(x) dx -> ∫' f(u) du

5.5, p. 415, Example 1

2 ways to evaluate Definate Integral:

1) Evaluate integral then use FTC2 (preferred)

2) Change limits of integration with substitution

(don’t resubs back in for u)

5.5, p. 418, Example 7

III) 5.6 Natural Logarithm as Integral, p. 422