Calculus 221, section 9.3 Definite Integrals
notes prepared by Tim Pilachowski
Back in Math 220, section 6.3, we made use of the Fundamental Theorem of Calculus:
Given a function f(x) which is continuous on an interval [a, b], and given F(x) which is an antiderivative of f(x),
then
() () () ()
aFbF
a
b
xFdxxf
b
a−==
∫.
Note that this theorem can make use of any antiderivative, so we’ll generally choose the easiest version, i.e. the
one without the “+ C”. This works because we’d always get “+ C – C = 0” anyway.
Example A: Evaluate
()
∫6
7
0
3cos
π
dtt . answer: 3
1
−
Example B: Find ∫2
12
ln
edx
x
x. answer: 1
Example B using the Change of Limits Rule: Find ∫2
12
ln
edx
x
x. answer: 1
Example C:
()
∫+
2
05
23
dx
x
x. answer: 0014911482.0
44 3
1
7
1
8
1≅
⎥
⎦
⎤
⎢
⎣
⎡−−
