. . . . . . Section 5.3 Evaluating Definite Integrals V63.0121, Calculus I April 20, 2009 Announcements I Final Exam is Friday, May 8, 2:00–3:50pm I Final is cumulative; topics will be represented roughly according to time spent on them . . Image credit: docman
Computing integrals with Riemann sums is like computing derivatives with limits. The calculus of integrals turns out to come from antidifferentiation. This startling fact is the Second Fundamental Theorem of Calculus!
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Section5.3EvaluatingDefiniteIntegrals
V63.0121, CalculusI
April20, 2009
Announcements
I FinalExamisFriday, May8, 2:00–3:50pmI Finaliscumulative; topicswillberepresentedroughlyaccordingtotimespentonthem
ex dx = ex + C∫sin x dx = − cos x + C∫cos x dx = sin x + C∫sec2 x dx = tan x + C∫
sec x tan x dx = sec x + C∫1
1 + x2dx = arctan x + C
∫cf(x)dx = c
∫f(x)dx∫
1xdx = ln |x| + C∫
ax dx =ax
ln a+ C∫
csc2 x dx = − cot x + C∫csc x cot x dx = − csc x + C∫
1√1− x2
dx = arcsin x + C
. . . . . .
Outline
Lasttime: TheDefiniteIntegral
EvaluatingDefiniteIntegralsExamples
TotalChange
IndefiniteIntegralsMyfirsttableofintegrals
Examples“NegativeArea”
. . . . . .
ExampleFindtheareabetweenthegraphof y = (x− 1)(x− 2), the x-axis,andtheverticallines x = 0 and x = 3.
Solution
Consider∫ 3
0(x− 1)(x− 2)dx. Noticetheintegrandispositiveon
[0, 1) and (2, 3], andnegativeon (1, 2). Ifwewanttheareaoftheregion, wehavetodo
A =
∫ 1
0(x− 1)(x− 2)dx−
∫ 2
1(x− 1)(x− 2)dx +
∫ 3
2(x− 1)(x− 2)dx
=[13x
3 − 32x
2 + 2x]10 −
[13x
3 − 32x
2 + 2x]21 +
[13x
3 − 32x
2 + 2x]32
=56−
(−16
)+
56
=116
.
. . . . . .
ExampleFindtheareabetweenthegraphof y = (x− 1)(x− 2), the x-axis,andtheverticallines x = 0 and x = 3.
Solution
Consider∫ 3
0(x− 1)(x− 2)dx. Noticetheintegrandispositiveon
[0, 1) and (2, 3], andnegativeon (1, 2). Ifwewanttheareaoftheregion, wehavetodo
A =
∫ 1
0(x− 1)(x− 2)dx−
∫ 2
1(x− 1)(x− 2)dx +
∫ 3
2(x− 1)(x− 2)dx
=[13x
3 − 32x
2 + 2x]10 −
[13x
3 − 32x
2 + 2x]21 +
[13x
3 − 32x
2 + 2x]32
=56−
(−16
)+
56
=116
.
. . . . . .
Graphfrompreviousexample
. .x
.y
..1
..2
..3
. . . . . .
Summary
I integralscanbecomputedwithantidifferentiationI integralofinstantaneousrateofchangeistotalnetchangeI ThesecondFunamentalTheoremofCalculusrequirestheMeanValueTheorem