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EECM 3714

Lecture 11: Unit 11

Integration
Renshaw, Ch
...
Also
known as the anti-derivative
...
r
...
x
• For indefinite integrals, always include the c
• Note the‫ )𝑥(𝑓 ׬‬just means integral sign f(x), while ‫ 𝑥𝑑 )𝑥(𝑓 ׬‬means that we have to integrate 𝑓(𝑥) w
...
t
...

• Notation is important here
...


• The derivative of the indefinite integral = the integrand
‫ 𝑥 𝐹 = 𝑥𝑑 )𝑥(𝑓 ׬‬+ 𝑐
...
5 𝑑𝑥

‫𝑥׬‬

0
...
5+1
0
...
5
1
...
5 + 𝑐
...
5
𝑥
𝑑𝑥 3

+ 𝑐 = 𝑥 0
...

0

• Also note that ‫ ׬ = 𝑥𝑑 ׬‬1 𝑑𝑥 = ‫= 𝑥𝑑 𝑥 ׬‬

𝑥 0+1
0+1

+𝑐 =𝑥+𝑐

MULTIPLICATIVE CONSTANT RULE
• Recall that:

𝑑
𝐴𝑥
𝑑𝑥

= 𝐴,

𝑑
𝐴𝑓
𝑑𝑥

𝑥 = 𝐴𝑓′(𝑥)

• To reverse this:

SUM/DIFFERENCE RULE
• Recall that

𝑑
𝑑𝑥

𝑓 𝑥 ± 𝑔(𝑥) = 𝑓′(𝑥) ± 𝑔′(𝑥)

‫ 𝑥 𝐹𝐴 = 𝑥𝑑 )𝑥(𝑓 ׬ 𝐴 = 𝑥𝑑 )𝑥(𝑓𝐴 ׬‬+ 𝑐

• To
reverse
this:
‫ 𝑥 𝑓 ׬‬± 𝑔(𝑥) 𝑑𝑥 =
‫ 𝑥𝑑 )𝑥(𝑓 ׬‬± ‫ 𝑥 𝐹 = 𝑥𝑑 )𝑥(𝑔 ׬‬± 𝐺 𝑥 + 𝑐

• Example: find ‫ ׬‬100𝑥 3 𝑑𝑥

• Example: find ‫ ׬‬5𝑥 3 + 3𝑥 2 𝑑𝑥

4
𝑥
න 100𝑥 3 𝑑𝑥 = 100 න 𝑥 3 𝑑𝑥 = 100
+𝑐
4

= 25𝑥 4 + 𝑐

‫ ׬‬5𝑥 3 + 3𝑥 2 𝑑𝑥 = ‫ ׬‬5𝑥 3 𝑑𝑥 + ‫ ׬‬3𝑥 2 𝑑𝑥 =
5 ‫ 𝑥 ׬‬3 𝑑𝑥 + 3 ‫ 𝑥 ׬‬2 𝑑𝑥
...
25𝑥 4 + 𝑥 3 + 𝑐
...
(this is just for you = 5
to check that you have integrated correctly)
• Derivative of this function is
...


• Recall that:

the

chain

rule

of

2

𝑑
𝑑𝑥

𝑥 3 + 3𝑥 = 3𝑥 2 + 3
3

• ∴ ‫ ׬‬3𝑥 + 3 𝑥 + 3𝑥
0
...


• Also recall that:

𝑑
ln
𝑑𝑥

𝑓(𝑥) =

• To reverse this:

‫׬‬

𝑓′(𝑥)
𝑓(𝑥)

𝑑𝑥 = ln 𝑓(𝑥) + 𝑐
...
10-11

𝑑𝑥 = 𝑒

𝑑𝑥 =

+𝑐

𝑒 𝑘𝑥
𝑘

+𝑐

EXAMPLES: LOGARITHMIC AND EXPONENTIAL FUNCTIONS
• Find ‫ ׬‬9𝑒 −3𝑥 𝑑𝑥
‫ ׬‬9𝑒

−3𝑥

• Find ‫׬‬
‫׬‬

5
𝑥

𝑑𝑥 = 9 ‫𝑒 ׬‬
5
𝑥

−3𝑥

𝑑𝑥 = 9

𝑒 −3𝑥
−3

+ 𝑐 = −3𝑒 −3𝑥 + 𝑐
...


• Find ‫ ׬‬3𝑥 2 𝑒 𝑥

3

𝑑𝑥

• Note that: 3𝑥 2 =
∴ ‫ ׬‬3𝑥 2 𝑒 𝑥
• Find ‫׬‬

3

𝑑
𝑑𝑥

𝑥3

3

𝑑𝑥 = 𝑒 𝑥 + 𝑐
...


DEFINITE INTEGRATION
• Definite integration - allows us to calculate the area under a function/curve (area between x-axis
and curve)
• We use the definite integral to find the area below a function/curve
...
18
...
3-18
...

• a is the lower limit of integration and b is the upper limit of integration
𝑏

• ‫ 𝑥𝑑 )𝑥(𝑓 𝑎׬‬then means the definite integral of 𝑓(𝑥) between limits a and b
• Note that with definite integration, the arbitrary constant falls away

FINDING A DEFINITE INTEGRAL
• Note that if ‫ 𝑥 𝐹 = 𝑥𝑑 )𝑥(𝑓 ׬‬+ 𝑐 then
𝑏

‫ 𝑏=𝑥| 𝑥 𝐹 = 𝑥𝑑 )𝑥(𝑓 𝑎׬‬− 𝐹 𝑥 |𝑥=𝑎
...


INTEGRATION - Economic Applications
• Derive TC from MC

• Derive TR from MR

• Consumer surplus
• Producer surplus



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